Ranalysis3.v

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Require Import Rbase.
Require Import Rfunctions.
Require Import Ranalysis1.
Require Import Ranalysis2.
Local Open Scope R_scope.

Division

Theorem derivable_pt_lim_div :
  forall (f1 f2:R -> R) (x l1 l2:R),
    derivable_pt_lim f1 x l1 ->
    derivable_pt_lim f2 x l2 ->
    f2 x <> 0 ->
    derivable_pt_lim (f1 / f2) x ((l1 * f2 x - l2 * f1 x) / Rsqr (f2 x)).forall (f1 f2 : R -> R) (x l1 l2 : R),
derivable_pt_lim f1 x l1 ->
derivable_pt_lim f2 x l2 ->
f2 x <> 0 ->
derivable_pt_lim (f1 / f2) x
  ((l1 * f2 x - l2 * f1 x) / (f2 x)²)
Proof.forall (f1 f2 : R -> R) (x l1 l2 : R),
derivable_pt_lim f1 x l1 ->
derivable_pt_lim f2 x l2 ->
f2 x <> 0 ->
derivable_pt_lim (f1 / f2) x
  ((l1 * f2 x - l2 * f1 x) / (f2 x)²)
  intros f1 f2 x l1 l2 H H0 H1.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
derivable_pt_lim (f1 / f2) x
  ((l1 * f2 x - l2 * f1 x) / (f2 x)²)
  cut (derivable_pt f2 x);
    [ intro X | unfold derivable_pt; exists l2; exact H0 ].f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
derivable_pt_lim (f1 / f2) x
  ((l1 * f2 x - l2 * f1 x) / (f2 x)²)
  assert (H2 := continuous_neq_0 _ _ (derivable_continuous_pt _ _ X) H1).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
H2:exists eps : posreal,
  forall h : R, Rabs h < eps -> f2 (x + h) <> 0
derivable_pt_lim (f1 / f2) x
  ((l1 * f2 x - l2 * f1 x) / (f2 x)²)
  elim H2; clear H2; intros eps_f2 H2.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
derivable_pt_lim (f1 / f2) x
  ((l1 * f2 x - l2 * f1 x) / (f2 x)²)
  unfold div_fct.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
derivable_pt_lim (fun x0 : R => f1 x0 / f2 x0) x
  ((l1 * f2 x - l2 * f1 x) / (f2 x)²)
  assert (H3 := derivable_continuous_pt _ _ X).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
H3:continuity_pt f2 x
derivable_pt_lim (fun x0 : R => f1 x0 / f2 x0) x
  ((l1 * f2 x - l2 * f1 x) / (f2 x)²)
  unfold continuity_pt in H3; unfold continue_in in H3; unfold limit1_in in H3;
    unfold limit_in in H3; unfold dist in H3.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
H3:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : Base R_met,
   D_x no_cond x x0 /\
   (let
      (Base, dist, _, _, _, _) as m
       return (Base m -> Base m -> R) := R_met in
    dist) x0 x < alp ->
   (let
      (Base, dist, _, _, _, _) as m
       return (Base m -> Base m -> R) := R_met in
    dist) (f2 x0) (f2 x) < eps)
derivable_pt_lim (fun x0 : R => f1 x0 / f2 x0) x
  ((l1 * f2 x - l2 * f1 x) / (f2 x)²)
  simpl in H3; unfold R_dist in H3.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
H3:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps)
derivable_pt_lim (fun x0 : R => f1 x0 / f2 x0) x
  ((l1 * f2 x - l2 * f1 x) / (f2 x)²)
  elim (H3 (Rabs (f2 x) / 2));
    [ idtac
      | unfold Rdiv; change (0 < Rabs (f2 x) * / 2);
        apply Rmult_lt_0_compat;
          [ apply Rabs_pos_lt; assumption | apply Rinv_0_lt_compat; prove_sup0 ] ].f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
H3:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps)
forall x0 : R,
x0 > 0 /\
(forall x1 : R,
 D_x no_cond x x1 /\ Rabs (x1 - x) < x0 ->
 Rabs (f2 x1 - f2 x) < Rabs (f2 x) / 2) ->
derivable_pt_lim (fun x1 : R => f1 x1 / f2 x1) x
  ((l1 * f2 x - l2 * f1 x) / (f2 x)²)
  clear H3; intros alp_f2 H3.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
derivable_pt_lim (fun x0 : R => f1 x0 / f2 x0) x
  ((l1 * f2 x - l2 * f1 x) / (f2 x)²)
  cut
    (forall x0:R,
      Rabs (x0 - x) < alp_f2 -> Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
(forall x0 : R,
 Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2) ->
derivable_pt_lim (fun x0 : R => f1 x0 / f2 x0) x
  ((l1 * f2 x - l2 * f1 x) / (f2 x)²)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intro H4.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
derivable_pt_lim (fun x0 : R => f1 x0 / f2 x0) x
  ((l1 * f2 x - l2 * f1 x) / (f2 x)²)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut (forall a:R, Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
(forall a : R,
 Rabs (a - x) < alp_f2 ->
 Rabs (f2 x) / 2 < Rabs (f2 a)) ->
derivable_pt_lim (fun x0 : R => f1 x0 / f2 x0) x
  ((l1 * f2 x - l2 * f1 x) / (f2 x)²)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intro H5.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
derivable_pt_lim (fun x0 : R => f1 x0 / f2 x0) x
  ((l1 * f2 x - l2 * f1 x) / (f2 x)²)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut
    (forall a:R,
      Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
(forall a : R,
 Rabs a < Rmin eps_f2 alp_f2 ->
 / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) ->
derivable_pt_lim (fun x0 : R => f1 x0 / f2 x0) x
  ((l1 * f2 x - l2 * f1 x) / (f2 x)²)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intro Maj.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
derivable_pt_lim (fun x0 : R => f1 x0 / f2 x0) x
  ((l1 * f2 x - l2 * f1 x) / (f2 x)²)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold derivable_pt_lim; intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  elim (H (Rabs (eps * f2 x / 8)));
    [ idtac
      | unfold Rdiv; change (0 < Rabs (eps * f2 x * / 8));
        apply Rabs_pos_lt; repeat apply prod_neq_R0;
          [ red; intro H7; rewrite H7 in H6; elim (Rlt_irrefl _ H6)
            | assumption
            | apply Rinv_neq_0_compat; discrR ] ].f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
forall x0 : posreal,
(forall h : R,
 h <> 0 ->
 Rabs h < x0 ->
 Rabs ((f1 (x + h) - f1 x) / h - l1) <
 Rabs (eps * f2 x / 8)) ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intros alp_f1d H7.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  case (Req_dec (f1 x) 0); intro.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  case (Req_dec l1 0); intro.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
(***********************************)
(*              First case         *)
(*           (f1 x)=0  l1 =0       *)
(***********************************)
  cut (0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d));
    [ intro
      | repeat apply Rmin_pos;
        [ apply (cond_pos eps_f2)
          | elim H3; intros; assumption
          | apply (cond_pos alp_f1d) ] ].f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  exists (mkposreal (Rmin eps_f2 (Rmin alp_f2 alp_f1d)) H10).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
forall h : R,
h <> 0 ->
Rabs h <
{|
pos := Rmin eps_f2 (Rmin alp_f2 alp_f1d);
cond_pos := H10 |} ->
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  simpl; intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H13 := Rlt_le_trans _ _ _ H12 (Rmin_r _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H14 := Rlt_le_trans _ _ _ H12 (Rmin_l _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H15 := Rlt_le_trans _ _ _ H13 (Rmin_r _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H16 := Rlt_le_trans _ _ _ H13 (Rmin_l _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H17 := H7 _ H11 H15).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite formule; [ idtac | assumption | assumption | apply H2; apply H14 ].f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs
  (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1) +
   l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h)) -
   f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2) +
   l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rle_lt_trans with
    (Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
      Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
      Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) +
      Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs
  (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1) +
   l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h)) -
   f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2) +
   l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) <=
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold Rminus.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs
  (/ f2 (x + h) * ((f1 (x + h) + - f1 x) / h + - l1) +
   l1 / (f2 x * f2 (x + h)) * (f2 x + - f2 (x + h)) +
   -
   (f1 x / (f2 x * f2 (x + h)) *
    ((f2 (x + h) + - f2 x) / h + - l2)) +
   l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) + - f2 x)) <=
Rabs
  (/ f2 (x + h) * ((f1 (x + h) + - f1 x) / h + - l1)) +
Rabs
  (l1 / (f2 x * f2 (x + h)) * (f2 x + - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) + - f2 x) / h + - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) + - f2 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite <-
    (Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2)))
    .f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs
  (/ f2 (x + h) * ((f1 (x + h) + - f1 x) / h + - l1) +
   l1 / (f2 x * f2 (x + h)) * (f2 x + - f2 (x + h)) +
   -
   (f1 x / (f2 x * f2 (x + h)) *
    ((f2 (x + h) + - f2 x) / h + - l2)) +
   l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) + - f2 x)) <=
Rabs
  (/ f2 (x + h) * ((f1 (x + h) + - f1 x) / h + - l1)) +
Rabs
  (l1 / (f2 x * f2 (x + h)) * (f2 x + - f2 (x + h))) +
Rabs
  (-
   (f1 x / (f2 x * f2 (x + h)) *
    ((f2 (x + h) + - f2 x) / h + - l2))) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) + - f2 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rabs_4.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  repeat rewrite Rabs_mult.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rlt_le_trans with (eps / 4 + eps / 4 + eps / 4 + eps / 4).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut (Rabs (/ f2 (x + h)) * Rabs ((f1 (x + h) - f1 x) / h - l1) < eps / 4).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut (Rabs (l1 / (f2 x * f2 (x + h))) * Rabs (f2 x - f2 (x + h)) < eps / 4).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut
    (Rabs (f1 x / (f2 x * f2 (x + h))) * Rabs ((f2 (x + h) - f2 x) / h - l2) <
      eps / 4).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4 ->
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut
    (Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h))) * Rabs (f2 (x + h) - f2 x) <
      eps / 4).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4 ->
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4 ->
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H18:Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4
H19:Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
H20:Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
H21:Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rlt_4; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite H8.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (l2 * 0 / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs 0 * Rabs (f2 (x + h) - f2 x) < eps * / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite Rabs_R0; rewrite Rmult_0_l.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
0 < eps * / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ].f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite H8.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (0 / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs 0 * Rabs ((f2 (x + h) - f2 x) * / h - l2) <
eps * / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite Rabs_R0; rewrite Rmult_0_l.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
0 < eps * / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ].f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite H9.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (0 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs 0 * Rabs (f2 x - f2 (x + h)) < eps * / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite Rabs_R0; rewrite Rmult_0_l.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
0 < eps * / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ].f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite <- Rabs_mult.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) <
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2);
    try assumption || apply H2.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs h < eps_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs h < Rmin eps_f2 alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply H14.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs h < Rmin eps_f2 alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rmin_2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 = 0
H10:0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
h:R
H11:h <> 0
H12:Rabs h < Rmin eps_f2 (Rmin alp_f2 alp_f1d)
H13:Rabs h < Rmin alp_f2 alp_f1d
H14:Rabs h < eps_f2
H15:Rabs h < alp_f1d
H16:Rabs h < alp_f2
H17:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  right; symmetry ; apply quadruple_var.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
(***********************************)
(*           Second case           *)
(*           (f1 x)=0  l1<>0       *)
(***********************************)
  assert (H10 := derivable_continuous_pt _ _ X).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:continuity_pt f2 x
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold continuity_pt in H10.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:continue_in f2 no_cond x
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold continue_in in H10.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:limit1_in f2 (D_x no_cond x) (f2 x) x
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold limit1_in in H10.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:limit_in R_met R_met f2 (D_x no_cond x) x (f2 x)
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold limit_in in H10.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : Base R_met,
   D_x no_cond x x0 /\ dist R_met x0 x < alp ->
   dist R_met (f2 x0) (f2 x) < eps0)
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold dist in H10.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : Base R_met,
   D_x no_cond x x0 /\
   (let
      (Base, dist, _, _, _, _) as m
       return (Base m -> Base m -> R) :=
      R_met in
    dist) x0 x < alp ->
   (let
      (Base, dist, _, _, _, _) as m
       return (Base m -> Base m -> R) :=
      R_met in
    dist) (f2 x0) (f2 x) < eps0)
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  simpl in H10.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ R_dist x0 x < alp ->
   R_dist (f2 x0) (f2 x) < eps0)
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold R_dist in H10.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  elim (H10 (Rabs (eps * Rsqr (f2 x) / (8 * l1)))).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
forall x0 : R,
x0 > 0 /\
(forall x1 : R,
 D_x no_cond x x1 /\ Rabs (x1 - x) < x0 ->
 Rabs (f2 x1 - f2 x) < Rabs (eps * (f2 x)² / (8 * l1))) ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  clear H10; intros alp_f2t2 H10.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut
    (forall a:R,
      Rabs a < alp_f2t2 ->
      Rabs (f2 (x + a) - f2 x) < Rabs (eps * Rsqr (f2 x) / (8 * l1))).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
(forall a : R,
 Rabs a < alp_f2t2 ->
 Rabs (f2 (x + a) - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1))) ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intro H11.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut (0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2) ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intro.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  exists (mkposreal (Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)) H12).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
forall h : R,
h <> 0 ->
Rabs h <
{|
pos := Rmin (Rmin eps_f2 alp_f1d)
         (Rmin alp_f2 alp_f2t2);
cond_pos := H12 |} ->
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  simpl.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
forall h : R,
h <> 0 ->
Rabs h <
Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2) ->
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H15 := Rlt_le_trans _ _ _ H14 (Rmin_r _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
H15:Rabs h < Rmin alp_f2 alp_f2t2
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H16 := Rlt_le_trans _ _ _ H14 (Rmin_l _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
H15:Rabs h < Rmin alp_f2 alp_f2t2
H16:Rabs h < Rmin eps_f2 alp_f1d
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H17 := Rlt_le_trans _ _ _ H15 (Rmin_l _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
H15:Rabs h < Rmin alp_f2 alp_f2t2
H16:Rabs h < Rmin eps_f2 alp_f1d
H17:Rabs h < alp_f2
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H18 := Rlt_le_trans _ _ _ H15 (Rmin_r _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
H15:Rabs h < Rmin alp_f2 alp_f2t2
H16:Rabs h < Rmin eps_f2 alp_f1d
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H19 := Rlt_le_trans _ _ _ H16 (Rmin_l _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
H15:Rabs h < Rmin alp_f2 alp_f2t2
H16:Rabs h < Rmin eps_f2 alp_f1d
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H20 := Rlt_le_trans _ _ _ H16 (Rmin_r _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
H15:Rabs h < Rmin alp_f2 alp_f2t2
H16:Rabs h < Rmin eps_f2 alp_f1d
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  clear H14 H15 H16.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite formule; try assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs
  (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1) +
   l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h)) -
   f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2) +
   l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rle_lt_trans with
    (Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
      Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
      Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) +
      Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs
  (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1) +
   l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h)) -
   f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2) +
   l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) <=
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold Rminus.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs
  (/ f2 (x + h) * ((f1 (x + h) + - f1 x) / h + - l1) +
   l1 / (f2 x * f2 (x + h)) * (f2 x + - f2 (x + h)) +
   -
   (f1 x / (f2 x * f2 (x + h)) *
    ((f2 (x + h) + - f2 x) / h + - l2)) +
   l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) + - f2 x)) <=
Rabs
  (/ f2 (x + h) * ((f1 (x + h) + - f1 x) / h + - l1)) +
Rabs
  (l1 / (f2 x * f2 (x + h)) * (f2 x + - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) + - f2 x) / h + - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) + - f2 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite <-
    (Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2)))
    .f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs
  (/ f2 (x + h) * ((f1 (x + h) + - f1 x) / h + - l1) +
   l1 / (f2 x * f2 (x + h)) * (f2 x + - f2 (x + h)) +
   -
   (f1 x / (f2 x * f2 (x + h)) *
    ((f2 (x + h) + - f2 x) / h + - l2)) +
   l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) + - f2 x)) <=
Rabs
  (/ f2 (x + h) * ((f1 (x + h) + - f1 x) / h + - l1)) +
Rabs
  (l1 / (f2 x * f2 (x + h)) * (f2 x + - f2 (x + h))) +
Rabs
  (-
   (f1 x / (f2 x * f2 (x + h)) *
    ((f2 (x + h) + - f2 x) / h + - l2))) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) + - f2 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rabs_4.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  repeat rewrite Rabs_mult.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rlt_le_trans with (eps / 4 + eps / 4 + eps / 4 + eps / 4).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut (Rabs (/ f2 (x + h)) * Rabs ((f1 (x + h) - f1 x) / h - l1) < eps / 4).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut (Rabs (l1 / (f2 x * f2 (x + h))) * Rabs (f2 x - f2 (x + h)) < eps / 4).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut
    (Rabs (f1 x / (f2 x * f2 (x + h))) * Rabs ((f2 (x + h) - f2 x) / h - l2) <
      eps / 4).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4 ->
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut
    (Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h))) * Rabs (f2 (x + h) - f2 x) <
      eps / 4).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4 ->
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4 ->
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
H14:Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4
H15:Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
H16:Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
H21:Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rlt_4; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite H8.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (l2 * 0 / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs 0 * Rabs (f2 (x + h) - f2 x) < eps * / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite Rabs_R0; rewrite Rmult_0_l.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
0 < eps * / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ].f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite H8.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (0 / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs 0 * Rabs ((f2 (x + h) - f2 x) * / h - l2) <
eps * / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite Rabs_R0; rewrite Rmult_0_l.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
0 < eps * / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ].f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite <- Rabs_mult.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) <
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply (maj_term2 x h eps l1 alp_f2 alp_f2t2 eps_f2 f2); try assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs h < Rmin eps_f2 alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply H2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs h < Rmin eps_f2 alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rmin_2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite <- Rabs_mult.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) <
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); try assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs h < Rmin eps_f2 alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply H2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
Rabs h < Rmin eps_f2 alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rmin_2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  right; symmetry ; apply quadruple_var.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H12:0 <
Rmin (Rmin eps_f2 alp_f1d)
  (Rmin alp_f2 alp_f2t2)
h:R
H13:h <> 0
H17:Rabs h < alp_f2
H18:Rabs h < alp_f2t2
H19:Rabs h < eps_f2
H20:Rabs h < alp_f1d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply H2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  repeat apply Rmin_pos.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < eps_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < alp_f1d
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < alp_f2t2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply (cond_pos eps_f2).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < alp_f1d
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < alp_f2t2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply (cond_pos alp_f1d).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < alp_f2t2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  elim H3; intros; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H11:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < alp_f2t2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  elim H10; intros; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
a:R
H11:Rabs a < alp_f2t2
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  elim H10; intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
a:R
H11:Rabs a < alp_f2t2
H12:alp_f2t2 > 0
H13:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  case (Req_dec a 0); intro.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
a:R
H11:Rabs a < alp_f2t2
H12:alp_f2t2 > 0
H13:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H14:a = 0
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
a:R
H11:Rabs a < alp_f2t2
H12:alp_f2t2 > 0
H13:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H14:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite H14; rewrite Rplus_0_r.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
a:R
H11:Rabs a < alp_f2t2
H12:alp_f2t2 > 0
H13:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H14:a = 0
Rabs (f2 x - f2 x) < Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
a:R
H11:Rabs a < alp_f2t2
H12:alp_f2t2 > 0
H13:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H14:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold Rminus; rewrite Rplus_opp_r.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
a:R
H11:Rabs a < alp_f2t2
H12:alp_f2t2 > 0
H13:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H14:a = 0
Rabs 0 < Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
a:R
H11:Rabs a < alp_f2t2
H12:alp_f2t2 > 0
H13:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H14:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite Rabs_R0.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
a:R
H11:Rabs a < alp_f2t2
H12:alp_f2t2 > 0
H13:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H14:a = 0
0 < Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
a:R
H11:Rabs a < alp_f2t2
H12:alp_f2t2 > 0
H13:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H14:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rabs_pos_lt.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
a:R
H11:Rabs a < alp_f2t2
H12:alp_f2t2 > 0
H13:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H14:a = 0
eps * (f2 x)² / (8 * l1) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
a:R
H11:Rabs a < alp_f2t2
H12:alp_f2t2 > 0
H13:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H14:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold Rdiv, Rsqr; repeat rewrite Rmult_assoc.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
a:R
H11:Rabs a < alp_f2t2
H12:alp_f2t2 > 0
H13:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H14:a = 0
eps * (f2 x * (f2 x * / (8 * l1))) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
a:R
H11:Rabs a < alp_f2t2
H12:alp_f2t2 > 0
H13:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H14:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  repeat apply prod_neq_R0; try assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
a:R
H11:Rabs a < alp_f2t2
H12:alp_f2t2 > 0
H13:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H14:a = 0
eps <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
a:R
H11:Rabs a < alp_f2t2
H12:alp_f2t2 > 0
H13:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H14:a = 0
/ (8 * l1) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
a:R
H11:Rabs a < alp_f2t2
H12:alp_f2t2 > 0
H13:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H14:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  now apply Rgt_not_eq.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
a:R
H11:Rabs a < alp_f2t2
H12:alp_f2t2 > 0
H13:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H14:a = 0
/ (8 * l1) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
a:R
H11:Rabs a < alp_f2t2
H12:alp_f2t2 > 0
H13:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H14:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rinv_neq_0_compat; apply prod_neq_R0; [discrR | assumption].f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
a:R
H11:Rabs a < alp_f2t2
H12:alp_f2t2 > 0
H13:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H14:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply H13.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
a:R
H11:Rabs a < alp_f2t2
H12:alp_f2t2 > 0
H13:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H14:a <> 0
D_x no_cond x (x + a) /\ Rabs (x + a - x) < alp_f2t2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  split.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
a:R
H11:Rabs a < alp_f2t2
H12:alp_f2t2 > 0
H13:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H14:a <> 0
D_x no_cond x (x + a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
a:R
H11:Rabs a < alp_f2t2
H12:alp_f2t2 > 0
H13:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H14:a <> 0
Rabs (x + a - x) < alp_f2t2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply D_x_no_cond; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
alp_f2t2:R
H10:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
a:R
H11:Rabs a < alp_f2t2
H12:alp_f2t2 > 0
H13:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H14:a <> 0
Rabs (x + a - x) < alp_f2t2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  replace (x + a - x) with a; [ assumption | ring ].f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  change (0 < Rabs (eps * Rsqr (f2 x) / (8 * l1))).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
0 < Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rabs_pos_lt; unfold Rdiv, Rsqr; repeat rewrite Rmult_assoc;
    repeat apply prod_neq_R0.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
eps <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
f2 x <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
f2 x <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
/ (8 * l1) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  red; intro; rewrite H11 in H6; elim (Rlt_irrefl _ H6).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
f2 x <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
f2 x <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
/ (8 * l1) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
f2 x <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
/ (8 * l1) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x = 0
H9:l1 <> 0
H10:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
/ (8 * l1) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rinv_neq_0_compat; apply prod_neq_R0; [discrR | assumption].f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
(***********************************)
(*        Third case               *)
(*     (f1 x)<>0  l1=0  l2=0       *)
(***********************************)
  case (Req_dec l1 0); intro.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  case (Req_dec l2 0); intro.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  elim (H0 (Rabs (Rsqr (f2 x) * eps / (8 * f1 x))));
    [ idtac
      | apply Rabs_pos_lt; unfold Rdiv, Rsqr; repeat rewrite Rmult_assoc;
        repeat apply prod_neq_R0 ;
          [ assumption
            | assumption
            | now apply Rgt_not_eq
            | apply Rinv_neq_0_compat; apply prod_neq_R0; discrR || assumption ] ].f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
forall x0 : posreal,
(forall h : R,
 h <> 0 ->
 Rabs h < x0 ->
 Rabs ((f2 (x + h) - f2 x) / h - l2) <
 Rabs ((f2 x)² * eps / (8 * f1 x))) ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intros alp_f2d H12.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut (0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d) ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intro.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  exists (mkposreal (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)) H11).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
forall h : R,
h <> 0 ->
Rabs h <
{|
pos := Rmin (Rmin eps_f2 alp_f2)
         (Rmin alp_f1d alp_f2d);
cond_pos := H11 |} ->
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  simpl.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
forall h : R,
h <> 0 ->
Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d) ->
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H15 := Rlt_le_trans _ _ _ H14 (Rmin_l _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H15:Rabs h < Rmin eps_f2 alp_f2
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H16 := Rlt_le_trans _ _ _ H14 (Rmin_r _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H15:Rabs h < Rmin eps_f2 alp_f2
H16:Rabs h < Rmin alp_f1d alp_f2d
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H17 := Rlt_le_trans _ _ _ H15 (Rmin_l _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H15:Rabs h < Rmin eps_f2 alp_f2
H16:Rabs h < Rmin alp_f1d alp_f2d
H17:Rabs h < eps_f2
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H18 := Rlt_le_trans _ _ _ H15 (Rmin_r _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H15:Rabs h < Rmin eps_f2 alp_f2
H16:Rabs h < Rmin alp_f1d alp_f2d
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H19 := Rlt_le_trans _ _ _ H16 (Rmin_l _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H15:Rabs h < Rmin eps_f2 alp_f2
H16:Rabs h < Rmin alp_f1d alp_f2d
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H20 := Rlt_le_trans _ _ _ H16 (Rmin_r _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H15:Rabs h < Rmin eps_f2 alp_f2
H16:Rabs h < Rmin alp_f1d alp_f2d
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  clear H15 H16.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite formule; try assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs
  (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1) +
   l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h)) -
   f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2) +
   l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rle_lt_trans with
    (Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
      Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
      Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) +
      Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs
  (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1) +
   l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h)) -
   f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2) +
   l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) <=
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold Rminus.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs
  (/ f2 (x + h) * ((f1 (x + h) + - f1 x) / h + - l1) +
   l1 / (f2 x * f2 (x + h)) * (f2 x + - f2 (x + h)) +
   -
   (f1 x / (f2 x * f2 (x + h)) *
    ((f2 (x + h) + - f2 x) / h + - l2)) +
   l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) + - f2 x)) <=
Rabs
  (/ f2 (x + h) * ((f1 (x + h) + - f1 x) / h + - l1)) +
Rabs
  (l1 / (f2 x * f2 (x + h)) * (f2 x + - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) + - f2 x) / h + - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) + - f2 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite <-
    (Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2)))
    .f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs
  (/ f2 (x + h) * ((f1 (x + h) + - f1 x) / h + - l1) +
   l1 / (f2 x * f2 (x + h)) * (f2 x + - f2 (x + h)) +
   -
   (f1 x / (f2 x * f2 (x + h)) *
    ((f2 (x + h) + - f2 x) / h + - l2)) +
   l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) + - f2 x)) <=
Rabs
  (/ f2 (x + h) * ((f1 (x + h) + - f1 x) / h + - l1)) +
Rabs
  (l1 / (f2 x * f2 (x + h)) * (f2 x + - f2 (x + h))) +
Rabs
  (-
   (f1 x / (f2 x * f2 (x + h)) *
    ((f2 (x + h) + - f2 x) / h + - l2))) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) + - f2 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rabs_4.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  repeat rewrite Rabs_mult.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rlt_le_trans with (eps / 4 + eps / 4 + eps / 4 + eps / 4).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut (Rabs (/ f2 (x + h)) * Rabs ((f1 (x + h) - f1 x) / h - l1) < eps / 4).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut (Rabs (l1 / (f2 x * f2 (x + h))) * Rabs (f2 x - f2 (x + h)) < eps / 4).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut
    (Rabs (f1 x / (f2 x * f2 (x + h))) * Rabs ((f2 (x + h) - f2 x) / h - l2) <
      eps / 4).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4 ->
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut
    (Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h))) * Rabs (f2 (x + h) - f2 x) <
      eps / 4).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4 ->
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4 ->
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
H15:Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4
H16:Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
H21:Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
H22:Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rlt_4; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite H10.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (0 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs 0 * Rabs (f2 (x + h) - f2 x) < eps * / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite Rabs_R0; rewrite Rmult_0_l.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
0 < eps * / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ].f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite <- Rabs_mult.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply (maj_term3 x h eps l2 alp_f2 eps_f2 alp_f2d f1 f2); try assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs h < Rmin eps_f2 alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply H2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs h < Rmin eps_f2 alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rmin_2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite H9.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (0 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs 0 * Rabs (f2 x - f2 (x + h)) < eps * / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite Rabs_R0; rewrite Rmult_0_l.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
0 < eps * / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ].f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite <- Rabs_mult.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) <
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); assumption || idtac.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs h < Rmin eps_f2 alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply H2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
Rabs h < Rmin eps_f2 alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rmin_2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  right; symmetry ; apply quadruple_var.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H11:0 <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
h:R
H13:h <> 0
H14:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H17:Rabs h < eps_f2
H18:Rabs h < alp_f2
H19:Rabs h < alp_f1d
H20:Rabs h < alp_f2d
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply H2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  repeat apply Rmin_pos.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < eps_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < alp_f1d
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < alp_f2d
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply (cond_pos eps_f2).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < alp_f1d
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < alp_f2d
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  elim H3; intros; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < alp_f1d
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < alp_f2d
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply (cond_pos alp_f1d).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 = 0
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < alp_f2d
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply (cond_pos alp_f2d).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
(***********************************)
(*      Fourth case                *)
(*    (f1 x)<>0  l1=0  l2<>0       *)
(***********************************)
  elim (H0 (Rabs (Rsqr (f2 x) * eps / (8 * f1 x))));
    [ idtac
      | apply Rabs_pos_lt; unfold Rsqr, Rdiv;
        repeat apply prod_neq_R0 ;
          [ assumption..
            | now apply Rgt_not_eq
            | apply Rinv_neq_0_compat; apply prod_neq_R0; discrR || assumption ] ].f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
forall x0 : posreal,
(forall h : R,
 h <> 0 ->
 Rabs h < x0 ->
 Rabs ((f2 (x + h) - f2 x) / h - l2) <
 Rabs ((f2 x)² * eps / (8 * f1 x))) ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intros alp_f2d H11.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H12 := derivable_continuous_pt _ _ X).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:continuity_pt f2 x
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold continuity_pt in H12.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:continue_in f2 no_cond x
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold continue_in in H12.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:limit1_in f2 (D_x no_cond x) (f2 x) x
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold limit1_in in H12.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:limit_in R_met R_met f2 (D_x no_cond x) x (f2 x)
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold limit_in in H12.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : Base R_met,
   D_x no_cond x x0 /\ dist R_met x0 x < alp ->
   dist R_met (f2 x0) (f2 x) < eps0)
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold dist in H12.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : Base R_met,
   D_x no_cond x x0 /\
   (let
      (Base, dist, _, _, _, _) as m
       return (Base m -> Base m -> R) :=
      R_met in
    dist) x0 x < alp ->
   (let
      (Base, dist, _, _, _, _) as m
       return (Base m -> Base m -> R) :=
      R_met in
    dist) (f2 x0) (f2 x) < eps0)
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  simpl in H12.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ R_dist x0 x < alp ->
   R_dist (f2 x0) (f2 x) < eps0)
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold R_dist in H12.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  elim (H12 (Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2)))).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
forall x0 : R,
x0 > 0 /\
(forall x1 : R,
 D_x no_cond x x1 /\ Rabs (x1 - x) < x0 ->
 Rabs (f2 x1 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))) ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intros alp_f2c H13.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut (0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d (Rmin alp_f2d alp_f2c))).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c)) ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intro.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  exists
    (mkposreal (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d (Rmin alp_f2d alp_f2c)))
      H14).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
forall h : R,
h <> 0 ->
Rabs h <
{|
pos := Rmin (Rmin eps_f2 alp_f2)
         (Rmin alp_f1d (Rmin alp_f2d alp_f2c));
cond_pos := H14 |} ->
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  simpl; intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H16:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H17 := Rlt_le_trans _ _ _ H16 (Rmin_l _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H16:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
H17:Rabs h < Rmin eps_f2 alp_f2
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H18 := Rlt_le_trans _ _ _ H16 (Rmin_r _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H16:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
H17:Rabs h < Rmin eps_f2 alp_f2
H18:Rabs h < Rmin alp_f1d (Rmin alp_f2d alp_f2c)
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H19 := Rlt_le_trans _ _ _ H18 (Rmin_r _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H16:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
H17:Rabs h < Rmin eps_f2 alp_f2
H18:Rabs h < Rmin alp_f1d (Rmin alp_f2d alp_f2c)
H19:Rabs h < Rmin alp_f2d alp_f2c
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H20 := Rlt_le_trans _ _ _ H19 (Rmin_l _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H16:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
H17:Rabs h < Rmin eps_f2 alp_f2
H18:Rabs h < Rmin alp_f1d (Rmin alp_f2d alp_f2c)
H19:Rabs h < Rmin alp_f2d alp_f2c
H20:Rabs h < alp_f2d
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H21 := Rlt_le_trans _ _ _ H19 (Rmin_r _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H16:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
H17:Rabs h < Rmin eps_f2 alp_f2
H18:Rabs h < Rmin alp_f1d (Rmin alp_f2d alp_f2c)
H19:Rabs h < Rmin alp_f2d alp_f2c
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H22 := Rlt_le_trans _ _ _ H18 (Rmin_l _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H16:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
H17:Rabs h < Rmin eps_f2 alp_f2
H18:Rabs h < Rmin alp_f1d (Rmin alp_f2d alp_f2c)
H19:Rabs h < Rmin alp_f2d alp_f2c
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H23 := Rlt_le_trans _ _ _ H17 (Rmin_l _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H16:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
H17:Rabs h < Rmin eps_f2 alp_f2
H18:Rabs h < Rmin alp_f1d (Rmin alp_f2d alp_f2c)
H19:Rabs h < Rmin alp_f2d alp_f2c
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H24 := Rlt_le_trans _ _ _ H17 (Rmin_r _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H16:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
H17:Rabs h < Rmin eps_f2 alp_f2
H18:Rabs h < Rmin alp_f1d (Rmin alp_f2d alp_f2c)
H19:Rabs h < Rmin alp_f2d alp_f2c
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  clear H16 H17 H18 H19.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut
    (forall a:R,
      Rabs a < alp_f2c ->
      Rabs (f2 (x + a) - f2 x) <
      Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
(forall a : R,
 Rabs a < alp_f2c ->
 Rabs (f2 (x + a) - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))) ->
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intro.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite formule; try assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs
  (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1) +
   l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h)) -
   f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2) +
   l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rle_lt_trans with
    (Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
      Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
      Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) +
      Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs
  (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1) +
   l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h)) -
   f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2) +
   l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) <=
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold Rminus.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs
  (/ f2 (x + h) * ((f1 (x + h) + - f1 x) / h + - l1) +
   l1 / (f2 x * f2 (x + h)) * (f2 x + - f2 (x + h)) +
   -
   (f1 x / (f2 x * f2 (x + h)) *
    ((f2 (x + h) + - f2 x) / h + - l2)) +
   l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) + - f2 x)) <=
Rabs
  (/ f2 (x + h) * ((f1 (x + h) + - f1 x) / h + - l1)) +
Rabs
  (l1 / (f2 x * f2 (x + h)) * (f2 x + - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) + - f2 x) / h + - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) + - f2 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite <-
    (Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2)))
    .f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs
  (/ f2 (x + h) * ((f1 (x + h) + - f1 x) / h + - l1) +
   l1 / (f2 x * f2 (x + h)) * (f2 x + - f2 (x + h)) +
   -
   (f1 x / (f2 x * f2 (x + h)) *
    ((f2 (x + h) + - f2 x) / h + - l2)) +
   l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) + - f2 x)) <=
Rabs
  (/ f2 (x + h) * ((f1 (x + h) + - f1 x) / h + - l1)) +
Rabs
  (l1 / (f2 x * f2 (x + h)) * (f2 x + - f2 (x + h))) +
Rabs
  (-
   (f1 x / (f2 x * f2 (x + h)) *
    ((f2 (x + h) + - f2 x) / h + - l2))) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) + - f2 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rabs_4.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  repeat rewrite Rabs_mult.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rlt_le_trans with (eps / 4 + eps / 4 + eps / 4 + eps / 4).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut (Rabs (/ f2 (x + h)) * Rabs ((f1 (x + h) - f1 x) / h - l1) < eps / 4).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut (Rabs (l1 / (f2 x * f2 (x + h))) * Rabs (f2 x - f2 (x + h)) < eps / 4).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut
    (Rabs (f1 x / (f2 x * f2 (x + h))) * Rabs ((f2 (x + h) - f2 x) / h - l2) <
      eps / 4).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4 ->
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut
    (Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h))) * Rabs (f2 (x + h) - f2 x) <
      eps / 4).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4 ->
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4 ->
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H17:Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4
H18:Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
H19:Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
H25:Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rlt_4; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite <- Rabs_mult.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply (maj_term4 x h eps l2 alp_f2 alp_f2c eps_f2 f1 f2); try assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs h < Rmin eps_f2 alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply H2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs h < Rmin eps_f2 alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rmin_2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite <- Rabs_mult.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply (maj_term3 x h eps l2 alp_f2 eps_f2 alp_f2d f1 f2); try assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs h < Rmin eps_f2 alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply H2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs h < Rmin eps_f2 alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rmin_2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite H9.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (0 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs 0 * Rabs (f2 x - f2 (x + h)) < eps * / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite Rabs_R0; rewrite Rmult_0_l.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
0 < eps * / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ].f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite <- Rabs_mult.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) <
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); try assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs h < Rmin eps_f2 alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply H2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs h < Rmin eps_f2 alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rmin_2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  right; symmetry ; apply quadruple_var.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
H16:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply H2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  case (Req_dec a 0); intro.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a = 0
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite H17; rewrite Rplus_0_r.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a = 0
Rabs (f2 x - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a = 0
0 < Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rabs_pos_lt.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a = 0
(f2 x)² * f2 x * eps / (8 * f1 x * l2) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold Rdiv, Rsqr.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a = 0
f2 x * f2 x * f2 x * eps * / (8 * f1 x * l2) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  repeat rewrite Rinv_mult_distr; try assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a = 0
f2 x * f2 x * f2 x * eps * (/ 8 * / f1 x * / l2) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a = 0
8 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a = 0
8 * f1 x <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  repeat apply prod_neq_R0; try assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a = 0
eps <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a = 0
/ 8 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a = 0
/ f1 x <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a = 0
/ l2 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a = 0
8 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a = 0
8 * f1 x <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  red; intro H18; rewrite H18 in H6; elim (Rlt_irrefl _ H6).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a = 0
/ 8 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a = 0
/ f1 x <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a = 0
/ l2 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a = 0
8 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a = 0
8 * f1 x <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rinv_neq_0_compat; discrR.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a = 0
/ f1 x <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a = 0
/ l2 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a = 0
8 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a = 0
8 * f1 x <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rinv_neq_0_compat; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a = 0
/ l2 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a = 0
8 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a = 0
8 * f1 x <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rinv_neq_0_compat; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a = 0
8 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a = 0
8 * f1 x <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  discrR.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a = 0
8 * f1 x <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply prod_neq_R0; [ discrR | assumption ].f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  elim H13; intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a <> 0
H18:alp_f2c > 0
H19:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
Rabs (f2 x0 - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply H19.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a <> 0
H18:alp_f2c > 0
H19:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
Rabs (f2 x0 - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
D_x no_cond x (x + a) /\ Rabs (x + a - x) < alp_f2c
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  split.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a <> 0
H18:alp_f2c > 0
H19:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
Rabs (f2 x0 - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
D_x no_cond x (x + a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a <> 0
H18:alp_f2c > 0
H19:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
Rabs (f2 x0 - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (x + a - x) < alp_f2c
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply D_x_no_cond; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
H14:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
h:R
H15:h <> 0
H20:Rabs h < alp_f2d
H21:Rabs h < alp_f2c
H22:Rabs h < alp_f1d
H23:Rabs h < eps_f2
H24:Rabs h < alp_f2
a:R
H16:Rabs a < alp_f2c
H17:a <> 0
H18:alp_f2c > 0
H19:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
Rabs (f2 x0 - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (x + a - x) < alp_f2c
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  replace (x + a - x) with a; [ assumption | ring ].f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2c))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  repeat apply Rmin_pos.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 < eps_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 < alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 < alp_f1d
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 < alp_f2d
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 < alp_f2c
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply (cond_pos eps_f2).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 < alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 < alp_f1d
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 < alp_f2d
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 < alp_f2c
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  elim H3; intros; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 < alp_f1d
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 < alp_f2d
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 < alp_f2c
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply (cond_pos alp_f1d).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 < alp_f2d
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 < alp_f2c
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply (cond_pos alp_f2d).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
0 < alp_f2c
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  elim H13; intros; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  change (0 < Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
0 < Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rabs_pos_lt.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
(f2 x)² * f2 x * eps / (8 * f1 x * l2) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold Rsqr, Rdiv.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
f2 x * f2 x * f2 x * eps * / (8 * f1 x * l2) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  repeat rewrite Rinv_mult_distr; try assumption || discrR.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
f2 x * f2 x * f2 x * eps * (/ 8 * / f1 x * / l2) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
8 * f1 x <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  repeat apply prod_neq_R0; try assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
eps <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
/ 8 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
/ f1 x <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
/ l2 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
8 * f1 x <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  red; intro H13; rewrite H13 in H6; elim (Rlt_irrefl _ H6).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
/ 8 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
/ f1 x <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
/ l2 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
8 * f1 x <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rinv_neq_0_compat; discrR.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
/ f1 x <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
/ l2 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
8 * f1 x <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rinv_neq_0_compat; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
/ l2 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
8 * f1 x <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rinv_neq_0_compat; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 = 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
8 * f1 x <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply prod_neq_R0; [ discrR | assumption ].f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
(***********************************)
(*         Fifth case              *)
(*    (f1 x)<>0  l1<>0  l2=0       *)
(***********************************)
  case (Req_dec l2 0); intro.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H11 := derivable_continuous_pt _ _ X).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:continuity_pt f2 x
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold continuity_pt in H11.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:continue_in f2 no_cond x
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold continue_in in H11.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:limit1_in f2 (D_x no_cond x) (f2 x) x
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold limit1_in in H11.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:limit_in R_met R_met f2 (D_x no_cond x) x (f2 x)
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold limit_in in H11.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : Base R_met,
   D_x no_cond x x0 /\ dist R_met x0 x < alp ->
   dist R_met (f2 x0) (f2 x) < eps0)
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold dist in H11.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : Base R_met,
   D_x no_cond x x0 /\
   (let
      (Base, dist, _, _, _, _) as m
       return (Base m -> Base m -> R) :=
      R_met in
    dist) x0 x < alp ->
   (let
      (Base, dist, _, _, _, _) as m
       return (Base m -> Base m -> R) :=
      R_met in
    dist) (f2 x0) (f2 x) < eps0)
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  simpl in H11.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ R_dist x0 x < alp ->
   R_dist (f2 x0) (f2 x) < eps0)
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold R_dist in H11.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  elim (H11 (Rabs (eps * Rsqr (f2 x) / (8 * l1)))).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
forall x0 : R,
x0 > 0 /\
(forall x1 : R,
 D_x no_cond x x1 /\ Rabs (x1 - x) < x0 ->
 Rabs (f2 x1 - f2 x) < Rabs (eps * (f2 x)² / (8 * l1))) ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  clear H11; intros alp_f2t2 H11.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  elim (H0 (Rabs (Rsqr (f2 x) * eps / (8 * f1 x)))).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
forall x0 : posreal,
(forall h : R,
 h <> 0 ->
 Rabs h < x0 ->
 Rabs ((f2 (x + h) - f2 x) / h - l2) <
 Rabs ((f2 x)² * eps / (8 * f1 x))) ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intros alp_f2d H12.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut (0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2)) ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intro.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  exists
    (mkposreal
      (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))) H13).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall h : R,
h <> 0 ->
Rabs h <
{|
pos := Rmin (Rmin eps_f2 alp_f2)
         (Rmin alp_f1d (Rmin alp_f2d alp_f2t2));
cond_pos := H13 |} ->
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  simpl.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall h : R,
h <> 0 ->
Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2)) ->
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut
    (forall a:R,
      Rabs a < alp_f2t2 ->
      Rabs (f2 (x + a) - f2 x) < Rabs (eps * Rsqr (f2 x) / (8 * l1))).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
(forall a : R,
 Rabs a < alp_f2t2 ->
 Rabs (f2 (x + a) - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1))) ->
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intro.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H17 := Rlt_le_trans _ _ _ H15 (Rmin_l _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H17:Rabs h < Rmin eps_f2 alp_f2
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H18 := Rlt_le_trans _ _ _ H15 (Rmin_r _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H17:Rabs h < Rmin eps_f2 alp_f2
H18:Rabs h < Rmin alp_f1d (Rmin alp_f2d alp_f2t2)
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H19 := Rlt_le_trans _ _ _ H17 (Rmin_r _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H17:Rabs h < Rmin eps_f2 alp_f2
H18:Rabs h < Rmin alp_f1d (Rmin alp_f2d alp_f2t2)
H19:Rabs h < alp_f2
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H20 := Rlt_le_trans _ _ _ H17 (Rmin_l _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H17:Rabs h < Rmin eps_f2 alp_f2
H18:Rabs h < Rmin alp_f1d (Rmin alp_f2d alp_f2t2)
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H21 := Rlt_le_trans _ _ _ H18 (Rmin_r _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H17:Rabs h < Rmin eps_f2 alp_f2
H18:Rabs h < Rmin alp_f1d (Rmin alp_f2d alp_f2t2)
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H21:Rabs h < Rmin alp_f2d alp_f2t2
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H22 := Rlt_le_trans _ _ _ H18 (Rmin_l _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H17:Rabs h < Rmin eps_f2 alp_f2
H18:Rabs h < Rmin alp_f1d (Rmin alp_f2d alp_f2t2)
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H21:Rabs h < Rmin alp_f2d alp_f2t2
H22:Rabs h < alp_f1d
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H23 := Rlt_le_trans _ _ _ H21 (Rmin_l _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H17:Rabs h < Rmin eps_f2 alp_f2
H18:Rabs h < Rmin alp_f1d (Rmin alp_f2d alp_f2t2)
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H21:Rabs h < Rmin alp_f2d alp_f2t2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H24 := Rlt_le_trans _ _ _ H21 (Rmin_r _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H17:Rabs h < Rmin eps_f2 alp_f2
H18:Rabs h < Rmin alp_f1d (Rmin alp_f2d alp_f2t2)
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H21:Rabs h < Rmin alp_f2d alp_f2t2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  clear H15 H17 H18 H21.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite formule; try assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs
  (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1) +
   l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h)) -
   f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2) +
   l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rle_lt_trans with
    (Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
      Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
      Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) +
      Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs
  (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1) +
   l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h)) -
   f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2) +
   l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) <=
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold Rminus.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs
  (/ f2 (x + h) * ((f1 (x + h) + - f1 x) / h + - l1) +
   l1 / (f2 x * f2 (x + h)) * (f2 x + - f2 (x + h)) +
   -
   (f1 x / (f2 x * f2 (x + h)) *
    ((f2 (x + h) + - f2 x) / h + - l2)) +
   l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) + - f2 x)) <=
Rabs
  (/ f2 (x + h) * ((f1 (x + h) + - f1 x) / h + - l1)) +
Rabs
  (l1 / (f2 x * f2 (x + h)) * (f2 x + - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) + - f2 x) / h + - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) + - f2 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite <-
    (Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2)))
    .f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs
  (/ f2 (x + h) * ((f1 (x + h) + - f1 x) / h + - l1) +
   l1 / (f2 x * f2 (x + h)) * (f2 x + - f2 (x + h)) +
   -
   (f1 x / (f2 x * f2 (x + h)) *
    ((f2 (x + h) + - f2 x) / h + - l2)) +
   l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) + - f2 x)) <=
Rabs
  (/ f2 (x + h) * ((f1 (x + h) + - f1 x) / h + - l1)) +
Rabs
  (l1 / (f2 x * f2 (x + h)) * (f2 x + - f2 (x + h))) +
Rabs
  (-
   (f1 x / (f2 x * f2 (x + h)) *
    ((f2 (x + h) + - f2 x) / h + - l2))) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) + - f2 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rabs_4.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  repeat rewrite Rabs_mult.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rlt_le_trans with (eps / 4 + eps / 4 + eps / 4 + eps / 4).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut (Rabs (/ f2 (x + h)) * Rabs ((f1 (x + h) - f1 x) / h - l1) < eps / 4).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut (Rabs (l1 / (f2 x * f2 (x + h))) * Rabs (f2 x - f2 (x + h)) < eps / 4).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut
    (Rabs (f1 x / (f2 x * f2 (x + h))) * Rabs ((f2 (x + h) - f2 x) / h - l2) <
      eps / 4).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4 ->
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut
    (Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h))) * Rabs (f2 (x + h) - f2 x) <
      eps / 4).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4 ->
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4 ->
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
H15:Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4
H17:Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
H18:Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
H21:Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rlt_4; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite H10.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (0 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs 0 * Rabs (f2 (x + h) - f2 x) < eps * / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite Rabs_R0; rewrite Rmult_0_l.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
0 < eps * / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ].f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite <- Rabs_mult.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply (maj_term3 x h eps l2 alp_f2 eps_f2 alp_f2d f1 f2); try assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs h < Rmin eps_f2 alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply H2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs h < Rmin eps_f2 alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rmin_2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite <- Rabs_mult.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) <
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply (maj_term2 x h eps l1 alp_f2 alp_f2t2 eps_f2 f2); try assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs h < Rmin eps_f2 alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply H2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs h < Rmin eps_f2 alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rmin_2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite <- Rabs_mult.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) <
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); try assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs h < Rmin eps_f2 alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply H2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
Rabs h < Rmin eps_f2 alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rmin_2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  right; symmetry ; apply quadruple_var.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H16:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs h < alp_f2
H20:Rabs h < eps_f2
H22:Rabs h < alp_f1d
H23:Rabs h < alp_f2d
H24:Rabs h < alp_f2t2
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply H2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
a:R
H16:Rabs a < alp_f2t2
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  case (Req_dec a 0); intro.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
a:R
H16:Rabs a < alp_f2t2
H17:a = 0
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
a:R
H16:Rabs a < alp_f2t2
H17:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite H17; rewrite Rplus_0_r; unfold Rminus; rewrite Rplus_opp_r;
    rewrite Rabs_R0.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
a:R
H16:Rabs a < alp_f2t2
H17:a = 0
0 < Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
a:R
H16:Rabs a < alp_f2t2
H17:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rabs_pos_lt.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
a:R
H16:Rabs a < alp_f2t2
H17:a = 0
eps * (f2 x)² / (8 * l1) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
a:R
H16:Rabs a < alp_f2t2
H17:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold Rdiv; rewrite Rinv_mult_distr; try discrR || assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
a:R
H16:Rabs a < alp_f2t2
H17:a = 0
eps * (f2 x)² * (/ 8 * / l1) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
a:R
H16:Rabs a < alp_f2t2
H17:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold Rsqr.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
a:R
H16:Rabs a < alp_f2t2
H17:a = 0
eps * (f2 x * f2 x) * (/ 8 * / l1) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
a:R
H16:Rabs a < alp_f2t2
H17:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  repeat apply prod_neq_R0;
    assumption ||
      (apply Rinv_neq_0_compat; assumption) ||
        (apply Rinv_neq_0_compat; discrR) ||
          (red; intro H18; rewrite H18 in H6; elim (Rlt_irrefl _ H6)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
a:R
H16:Rabs a < alp_f2t2
H17:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  elim H11; intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
a:R
H16:Rabs a < alp_f2t2
H17:a <> 0
H18:alp_f2t2 > 0
H19:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply H19.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
a:R
H16:Rabs a < alp_f2t2
H17:a <> 0
H18:alp_f2t2 > 0
H19:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
D_x no_cond x (x + a) /\ Rabs (x + a - x) < alp_f2t2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  split.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
a:R
H16:Rabs a < alp_f2t2
H17:a <> 0
H18:alp_f2t2 > 0
H19:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
D_x no_cond x (x + a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
a:R
H16:Rabs a < alp_f2t2
H17:a <> 0
H18:alp_f2t2 > 0
H19:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (x + a - x) < alp_f2t2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply D_x_no_cond; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H13:0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
h:R
H14:h <> 0
H15:Rabs h <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
a:R
H16:Rabs a < alp_f2t2
H17:a <> 0
H18:alp_f2t2 > 0
H19:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (x + a - x) < alp_f2t2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  replace (x + a - x) with a; [ assumption | ring ].f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 <
Rmin (Rmin eps_f2 alp_f2)
  (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  repeat apply Rmin_pos.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < eps_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < alp_f1d
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < alp_f2d
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < alp_f2t2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply (cond_pos eps_f2).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < alp_f1d
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < alp_f2d
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < alp_f2t2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  elim H3; intros; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < alp_f1d
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < alp_f2d
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < alp_f2t2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply (cond_pos alp_f1d).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < alp_f2d
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < alp_f2t2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply (cond_pos alp_f2d).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
alp_f2d:posreal
H12:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
0 < alp_f2t2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  elim H11; intros; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rabs_pos_lt.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
(f2 x)² * eps / (8 * f1 x) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold Rdiv, Rsqr; rewrite Rinv_mult_distr; try discrR || assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
alp_f2t2:R
H11:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
f2 x * f2 x * eps * (/ 8 * / f1 x) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  repeat apply prod_neq_R0;
    assumption ||
      (apply Rinv_neq_0_compat; assumption) ||
        (apply Rinv_neq_0_compat; discrR) ||
          (red; intro H12; rewrite H12 in H6; elim (Rlt_irrefl _ H6)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  change (0 < Rabs (eps * Rsqr (f2 x) / (8 * l1))).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
0 < Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rabs_pos_lt.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
eps * (f2 x)² / (8 * l1) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold Rdiv, Rsqr; rewrite Rinv_mult_distr; try discrR || assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 = 0
H11:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
eps * (f2 x * f2 x) * (/ 8 * / l1) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  repeat apply prod_neq_R0;
    assumption ||
      (apply Rinv_neq_0_compat; assumption) ||
        (apply Rinv_neq_0_compat; discrR) ||
          (red; intro H12; rewrite H12 in H6; elim (Rlt_irrefl _ H6)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
(***********************************)
(*       Sixth case                *)
(*    (f1 x)<>0  l1<>0  l2<>0      *)
(***********************************)
  elim (H0 (Rabs (Rsqr (f2 x) * eps / (8 * f1 x)))).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
forall x0 : posreal,
(forall h : R,
 h <> 0 ->
 Rabs h < x0 ->
 Rabs ((f2 (x + h) - f2 x) / h - l2) <
 Rabs ((f2 x)² * eps / (8 * f1 x))) ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intros alp_f2d H11.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H12 := derivable_continuous_pt _ _ X).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:continuity_pt f2 x
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold continuity_pt in H12.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:continue_in f2 no_cond x
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold continue_in in H12.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:limit1_in f2 (D_x no_cond x) (f2 x) x
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold limit1_in in H12.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:limit_in R_met R_met f2 (D_x no_cond x) x (f2 x)
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold limit_in in H12.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : Base R_met,
   D_x no_cond x x0 /\ dist R_met x0 x < alp ->
   dist R_met (f2 x0) (f2 x) < eps0)
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold dist in H12.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : Base R_met,
   D_x no_cond x x0 /\
   (let
      (Base, dist, _, _, _, _) as m
       return (Base m -> Base m -> R) :=
      R_met in
    dist) x0 x < alp ->
   (let
      (Base, dist, _, _, _, _) as m
       return (Base m -> Base m -> R) :=
      R_met in
    dist) (f2 x0) (f2 x) < eps0)
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  simpl in H12.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ R_dist x0 x < alp ->
   R_dist (f2 x0) (f2 x) < eps0)
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold R_dist in H12.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  elim (H12 (Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2)))).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
forall x0 : R,
x0 > 0 /\
(forall x1 : R,
 D_x no_cond x x1 /\ Rabs (x1 - x) < x0 ->
 Rabs (f2 x1 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))) ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intros alp_f2c H13.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  elim (H12 (Rabs (eps * Rsqr (f2 x) / (8 * l1)))).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
forall x0 : R,
x0 > 0 /\
(forall x1 : R,
 D_x no_cond x x1 /\ Rabs (x1 - x) < x0 ->
 Rabs (f2 x1 - f2 x) < Rabs (eps * (f2 x)² / (8 * l1))) ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intros alp_f2t2 H14.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut
    (0 <
      Rmin (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
      (Rmin alp_f2c alp_f2t2)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2) ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intro.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
     (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  exists
    (mkposreal
      (Rmin (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
        (Rmin alp_f2c alp_f2t2)) H15).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
forall h : R,
h <> 0 ->
Rabs h <
{|
pos := Rmin
         (Rmin (Rmin eps_f2 alp_f2)
            (Rmin alp_f1d alp_f2d))
         (Rmin alp_f2c alp_f2t2);
cond_pos := H15 |} ->
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  simpl.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
forall h : R,
h <> 0 ->
Rabs h <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2) ->
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H17:Rabs h <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H18 := Rlt_le_trans _ _ _ H17 (Rmin_l _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H17:Rabs h <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
H18:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H19 := Rlt_le_trans _ _ _ H17 (Rmin_r _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H17:Rabs h <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
H18:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H19:Rabs h < Rmin alp_f2c alp_f2t2
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H20 := Rlt_le_trans _ _ _ H18 (Rmin_l _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H17:Rabs h <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
H18:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H19:Rabs h < Rmin alp_f2c alp_f2t2
H20:Rabs h < Rmin eps_f2 alp_f2
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H21 := Rlt_le_trans _ _ _ H18 (Rmin_r _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H17:Rabs h <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
H18:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H19:Rabs h < Rmin alp_f2c alp_f2t2
H20:Rabs h < Rmin eps_f2 alp_f2
H21:Rabs h < Rmin alp_f1d alp_f2d
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H22 := Rlt_le_trans _ _ _ H19 (Rmin_l _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H17:Rabs h <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
H18:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H19:Rabs h < Rmin alp_f2c alp_f2t2
H20:Rabs h < Rmin eps_f2 alp_f2
H21:Rabs h < Rmin alp_f1d alp_f2d
H22:Rabs h < alp_f2c
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H23 := Rlt_le_trans _ _ _ H19 (Rmin_r _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H17:Rabs h <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
H18:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H19:Rabs h < Rmin alp_f2c alp_f2t2
H20:Rabs h < Rmin eps_f2 alp_f2
H21:Rabs h < Rmin alp_f1d alp_f2d
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H24 := Rlt_le_trans _ _ _ H20 (Rmin_l _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H17:Rabs h <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
H18:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H19:Rabs h < Rmin alp_f2c alp_f2t2
H20:Rabs h < Rmin eps_f2 alp_f2
H21:Rabs h < Rmin alp_f1d alp_f2d
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H25 := Rlt_le_trans _ _ _ H20 (Rmin_r _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H17:Rabs h <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
H18:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H19:Rabs h < Rmin alp_f2c alp_f2t2
H20:Rabs h < Rmin eps_f2 alp_f2
H21:Rabs h < Rmin alp_f1d alp_f2d
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H26 := Rlt_le_trans _ _ _ H21 (Rmin_l _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H17:Rabs h <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
H18:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H19:Rabs h < Rmin alp_f2c alp_f2t2
H20:Rabs h < Rmin eps_f2 alp_f2
H21:Rabs h < Rmin alp_f1d alp_f2d
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H27 := Rlt_le_trans _ _ _ H21 (Rmin_r _ _)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H17:Rabs h <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
H18:Rabs h <
Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)
H19:Rabs h < Rmin alp_f2c alp_f2t2
H20:Rabs h < Rmin eps_f2 alp_f2
H21:Rabs h < Rmin alp_f1d alp_f2d
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  clear H17 H18 H19 H20 H21.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut
    (forall a:R,
      Rabs a < alp_f2t2 ->
      Rabs (f2 (x + a) - f2 x) < Rabs (eps * Rsqr (f2 x) / (8 * l1))).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
(forall a : R,
 Rabs a < alp_f2t2 ->
 Rabs (f2 (x + a) - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1))) ->
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut
    (forall a:R,
      Rabs a < alp_f2c ->
      Rabs (f2 (x + a) - f2 x) <
      Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
(forall a : R,
 Rabs a < alp_f2c ->
 Rabs (f2 (x + a) - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))) ->
(forall a : R,
 Rabs a < alp_f2t2 ->
 Rabs (f2 (x + a) - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1))) ->
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs
  ((f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / (f2 x)²) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite formule; try assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs
  (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1) +
   l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h)) -
   f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2) +
   l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rle_lt_trans with
    (Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
      Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
      Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) +
      Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs
  (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1) +
   l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h)) -
   f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2) +
   l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) <=
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold Rminus.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs
  (/ f2 (x + h) * ((f1 (x + h) + - f1 x) / h + - l1) +
   l1 / (f2 x * f2 (x + h)) * (f2 x + - f2 (x + h)) +
   -
   (f1 x / (f2 x * f2 (x + h)) *
    ((f2 (x + h) + - f2 x) / h + - l2)) +
   l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) + - f2 x)) <=
Rabs
  (/ f2 (x + h) * ((f1 (x + h) + - f1 x) / h + - l1)) +
Rabs
  (l1 / (f2 x * f2 (x + h)) * (f2 x + - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) + - f2 x) / h + - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) + - f2 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite <-
    (Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2)))
    .f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs
  (/ f2 (x + h) * ((f1 (x + h) + - f1 x) / h + - l1) +
   l1 / (f2 x * f2 (x + h)) * (f2 x + - f2 (x + h)) +
   -
   (f1 x / (f2 x * f2 (x + h)) *
    ((f2 (x + h) + - f2 x) / h + - l2)) +
   l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) + - f2 x)) <=
Rabs
  (/ f2 (x + h) * ((f1 (x + h) + - f1 x) / h + - l1)) +
Rabs
  (l1 / (f2 x * f2 (x + h)) * (f2 x + - f2 (x + h))) +
Rabs
  (-
   (f1 x / (f2 x * f2 (x + h)) *
    ((f2 (x + h) + - f2 x) / h + - l2))) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) + - f2 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rabs_4.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) +
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  repeat rewrite Rabs_mult.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rlt_le_trans with (eps / 4 + eps / 4 + eps / 4 + eps / 4).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut (Rabs (/ f2 (x + h)) * Rabs ((f1 (x + h) - f1 x) / h - l1) < eps / 4).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut (Rabs (l1 / (f2 x * f2 (x + h))) * Rabs (f2 x - f2 (x + h)) < eps / 4).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut
    (Rabs (f1 x / (f2 x * f2 (x + h))) * Rabs ((f2 (x + h) - f2 x) / h - l2) <
      eps / 4).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4 ->
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  cut
    (Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h))) * Rabs (f2 (x + h) - f2 x) <
      eps / 4).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4 ->
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4 ->
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4 ->
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H19:Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4
H20:Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
H21:Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
H28:Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) +
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) +
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) +
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <
eps / 4 + eps / 4 + eps / 4 + eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rlt_4; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite <- Rabs_mult.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply (maj_term4 x h eps l2 alp_f2 alp_f2c eps_f2 f1 f2); try assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs h < Rmin eps_f2 alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply H2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs h < Rmin eps_f2 alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rmin_2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite <- Rabs_mult.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply (maj_term3 x h eps l2 alp_f2 eps_f2 alp_f2d f1 f2); try assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs h < Rmin eps_f2 alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply H2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs h < Rmin eps_f2 alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rmin_2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (f2 x - f2 (x + h)) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite <- Rabs_mult.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) <
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply (maj_term2 x h eps l1 alp_f2 alp_f2t2 eps_f2 f2); try assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs h < Rmin eps_f2 alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply H2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs h < Rmin eps_f2 alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rmin_2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (/ f2 (x + h)) *
Rabs ((f1 (x + h) - f1 x) / h - l1) < 
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite <- Rabs_mult.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) <
eps / 4
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); try assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs h < Rmin eps_f2 alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply H2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs h < Rmin eps_f2 alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rmin_2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps / 4 + eps / 4 + eps / 4 + eps / 4 <= eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  right; symmetry ; apply quadruple_var.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
H17:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H18:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f2 (x + h) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply H2; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2c
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  case (Req_dec a 0); intro.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2c
H18:a = 0
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2c
H18:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite H18; rewrite Rplus_0_r; unfold Rminus; rewrite Rplus_opp_r;
    rewrite Rabs_R0; apply Rabs_pos_lt.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2c
H18:a = 0
(f2 x)² * f2 x * eps / (8 * f1 x * l2) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2c
H18:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold Rdiv, Rsqr; rewrite Rinv_mult_distr.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2c
H18:a = 0
f2 x * f2 x * f2 x * eps * (/ (8 * f1 x) * / l2) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2c
H18:a = 0
8 * f1 x <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2c
H18:a = 0
l2 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2c
H18:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  repeat apply prod_neq_R0;
    assumption ||
      (apply Rinv_neq_0_compat; assumption) ||
        (apply Rinv_neq_0_compat; discrR) ||
          (red; intro H28; rewrite H28 in H6; elim (Rlt_irrefl _ H6)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2c
H18:a = 0
8 * f1 x <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2c
H18:a = 0
8 * f1 x <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2c
H18:a = 0
l2 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2c
H18:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply prod_neq_R0; [ discrR | assumption ].f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2c
H18:a = 0
8 * f1 x <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2c
H18:a = 0
l2 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2c
H18:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply prod_neq_R0; [ discrR | assumption ].f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2c
H18:a = 0
l2 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2c
H18:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2c
H18:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  elim H13; intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2c
H18:a <> 0
H19:alp_f2c > 0
H20:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
Rabs (f2 x0 - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply H20.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2c
H18:a <> 0
H19:alp_f2c > 0
H20:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
Rabs (f2 x0 - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
D_x no_cond x (x + a) /\ Rabs (x + a - x) < alp_f2c
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  split.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2c
H18:a <> 0
H19:alp_f2c > 0
H20:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
Rabs (f2 x0 - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
D_x no_cond x (x + a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2c
H18:a <> 0
H19:alp_f2c > 0
H20:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
Rabs (f2 x0 - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (x + a - x) < alp_f2c
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply D_x_no_cond; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2c
H18:a <> 0
H19:alp_f2c > 0
H20:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
Rabs (f2 x0 - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
Rabs (x + a - x) < alp_f2c
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  replace (x + a - x) with a; [ assumption | ring ].f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2t2
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  case (Req_dec a 0); intro.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2t2
H18:a = 0
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2t2
H18:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite H18; rewrite Rplus_0_r; unfold Rminus; rewrite Rplus_opp_r;
    rewrite Rabs_R0; apply Rabs_pos_lt.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2t2
H18:a = 0
eps * (f2 x)² / (8 * l1) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2t2
H18:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold Rdiv, Rsqr; rewrite Rinv_mult_distr.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2t2
H18:a = 0
eps * (f2 x * f2 x) * (/ 8 * / l1) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2t2
H18:a = 0
8 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2t2
H18:a = 0
l1 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2t2
H18:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  repeat apply prod_neq_R0;
    assumption ||
      (apply Rinv_neq_0_compat; assumption) ||
        (apply Rinv_neq_0_compat; discrR) ||
          (red; intro H28; rewrite H28 in H6; elim (Rlt_irrefl _ H6)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2t2
H18:a = 0
8 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2t2
H18:a = 0
l1 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2t2
H18:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  discrR.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2t2
H18:a = 0
l1 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2t2
H18:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2t2
H18:a <> 0
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  elim H14; intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2t2
H18:a <> 0
H19:alp_f2t2 > 0
H20:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply H20.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2t2
H18:a <> 0
H19:alp_f2t2 > 0
H20:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
D_x no_cond x (x + a) /\ Rabs (x + a - x) < alp_f2t2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  split.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2t2
H18:a <> 0
H19:alp_f2t2 > 0
H20:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
D_x no_cond x (x + a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2t2
H18:a <> 0
H19:alp_f2t2 > 0
H20:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (x + a - x) < alp_f2t2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold D_x, no_cond; split.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2t2
H18:a <> 0
H19:alp_f2t2 > 0
H20:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
True
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2t2
H18:a <> 0
H19:alp_f2t2 > 0
H20:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
x <> x + a
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2t2
H18:a <> 0
H19:alp_f2t2 > 0
H20:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (x + a - x) < alp_f2t2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  trivial.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2t2
H18:a <> 0
H19:alp_f2t2 > 0
H20:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
x <> x + a
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2t2
H18:a <> 0
H19:alp_f2t2 > 0
H20:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (x + a - x) < alp_f2t2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rminus_not_eq_right.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2t2
H18:a <> 0
H19:alp_f2t2 > 0
H20:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
x + a - x <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2t2
H18:a <> 0
H19:alp_f2t2 > 0
H20:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (x + a - x) < alp_f2t2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  replace (x + a - x) with a; [ assumption | ring ].f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h0 : R,
Rabs h0 < eps_f2 -> f2 (x + h0) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
Maj:forall a0 : R,
Rabs a0 < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a0)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
H15:0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2)
     (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
h:R
H16:h <> 0
H22:Rabs h < alp_f2c
H23:Rabs h < alp_f2t2
H24:Rabs h < eps_f2
H25:Rabs h < alp_f2
H26:Rabs h < alp_f1d
H27:Rabs h < alp_f2d
a:R
H17:Rabs a < alp_f2t2
H18:a <> 0
H19:alp_f2t2 > 0
H20:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
Rabs (f2 x0 - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (x + a - x) < alp_f2t2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  replace (x + a - x) with a; [ assumption | ring ].f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 <
Rmin
  (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
  (Rmin alp_f2c alp_f2t2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  repeat apply Rmin_pos.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < eps_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < alp_f1d
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < alp_f2d
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < alp_f2c
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < alp_f2t2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply (cond_pos eps_f2).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < alp_f1d
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < alp_f2d
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < alp_f2c
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < alp_f2t2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  elim H3; intros; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < alp_f1d
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < alp_f2d
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < alp_f2c
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < alp_f2t2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply (cond_pos alp_f1d).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < alp_f2d
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < alp_f2c
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < alp_f2t2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply (cond_pos alp_f2d).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < alp_f2c
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < alp_f2t2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  elim H13; intros; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
alp_f2t2:R
H14:alp_f2t2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2t2 ->
 Rabs (f2 x0 - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1)))
0 < alp_f2t2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  elim H14; intros; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
Rabs (eps * (f2 x)² / (8 * l1)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  change (0 < Rabs (eps * Rsqr (f2 x) / (8 * l1))); apply Rabs_pos_lt.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
eps * (f2 x)² / (8 * l1) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold Rdiv, Rsqr; rewrite Rinv_mult_distr; try discrR || assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
alp_f2c:R
H13:alp_f2c > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2c ->
 Rabs (f2 x0 - f2 x) <
 Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)))
eps * (f2 x * f2 x) * (/ 8 * / l1) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  repeat apply prod_neq_R0;
    assumption ||
      (apply Rinv_neq_0_compat; assumption) ||
        (apply Rinv_neq_0_compat; discrR) ||
          (red; intro H14; rewrite H14 in H6; elim (Rlt_irrefl _ H6)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  change (0 < Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2)));
    apply Rabs_pos_lt.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
(f2 x)² * f2 x * eps / (8 * f1 x * l2) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold Rdiv, Rsqr; rewrite Rinv_mult_distr.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
f2 x * f2 x * f2 x * eps * (/ (8 * f1 x) * / l2) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
8 * f1 x <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
l2 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  repeat apply prod_neq_R0;
    assumption ||
      (apply Rinv_neq_0_compat; assumption) ||
        (apply Rinv_neq_0_compat; discrR) ||
          (red; intro H13; rewrite H13 in H6; elim (Rlt_irrefl _ H6)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
8 * f1 x <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
8 * f1 x <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
l2 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply prod_neq_R0; [ discrR | assumption ].f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
8 * f1 x <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
l2 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply prod_neq_R0; [ discrR | assumption ].f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
alp_f2d:posreal
H11:forall h : R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H12:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 x0 - f2 x) < eps0)
l2 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rabs_pos_lt.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
(f2 x)² * eps / (8 * f1 x) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold Rdiv, Rsqr; rewrite Rinv_mult_distr;
    [ idtac | discrR | assumption ].f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
Maj:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
eps:R
H6:0 < eps
alp_f1d:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
H8:f1 x <> 0
H9:l1 <> 0
H10:l2 <> 0
f2 x * f2 x * eps * (/ 8 * / f1 x) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  repeat apply prod_neq_R0;
    assumption ||
      (apply Rinv_neq_0_compat; assumption) ||
        (apply Rinv_neq_0_compat; discrR) ||
          (red; intro H11; rewrite H11 in H6; elim (Rlt_irrefl _ H6)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a : R,
Rabs (a - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a)
forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold Rdiv.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
/ Rabs (f2 (x + a)) < 2 * / Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rmult_lt_reg_l with (Rabs (f2 (x + a))).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
0 < Rabs (f2 (x + a))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 (x + a)) * / Rabs (f2 (x + a)) <
Rabs (f2 (x + a)) * (2 * / Rabs (f2 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rabs_pos_lt; apply H2.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs a < eps_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 (x + a)) * / Rabs (f2 (x + a)) <
Rabs (f2 (x + a)) * (2 * / Rabs (f2 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rlt_le_trans with (Rmin eps_f2 alp_f2).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs a < Rmin eps_f2 alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rmin eps_f2 alp_f2 <= eps_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 (x + a)) * / Rabs (f2 (x + a)) <
Rabs (f2 (x + a)) * (2 * / Rabs (f2 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rmin eps_f2 alp_f2 <= eps_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 (x + a)) * / Rabs (f2 (x + a)) <
Rabs (f2 (x + a)) * (2 * / Rabs (f2 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rmin_l.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 (x + a)) * / Rabs (f2 (x + a)) <
Rabs (f2 (x + a)) * (2 * / Rabs (f2 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite <- Rinv_r_sym.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
1 < Rabs (f2 (x + a)) * (2 * / Rabs (f2 x))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 (x + a)) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rmult_lt_reg_l with (Rabs (f2 x)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
0 < Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 x) * 1 <
Rabs (f2 x) *
(Rabs (f2 (x + a)) * (2 * / Rabs (f2 x)))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 (x + a)) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rabs_pos_lt; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 x) * 1 <
Rabs (f2 x) *
(Rabs (f2 (x + a)) * (2 * / Rabs (f2 x)))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 (x + a)) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite Rmult_1_r.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 x) <
Rabs (f2 x) *
(Rabs (f2 (x + a)) * (2 * / Rabs (f2 x)))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 (x + a)) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite (Rmult_comm (Rabs (f2 x))).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 x) <
Rabs (f2 (x + a)) * (2 * / Rabs (f2 x)) * Rabs (f2 x)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 (x + a)) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  repeat rewrite Rmult_assoc.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 x) <
Rabs (f2 (x + a)) *
(2 * (/ Rabs (f2 x) * Rabs (f2 x)))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 (x + a)) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite <- Rinv_l_sym.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 x) < Rabs (f2 (x + a)) * (2 * 1)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 x) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 (x + a)) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite Rmult_1_r.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 x) < Rabs (f2 (x + a)) * 2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 x) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 (x + a)) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rmult_lt_reg_l with (/ 2).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
0 < / 2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
/ 2 * Rabs (f2 x) < / 2 * (Rabs (f2 (x + a)) * 2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 x) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 (x + a)) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rinv_0_lt_compat; prove_sup0.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
/ 2 * Rabs (f2 x) < / 2 * (Rabs (f2 (x + a)) * 2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 x) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 (x + a)) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  repeat rewrite (Rmult_comm (/ 2)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 x) * / 2 < Rabs (f2 (x + a)) * 2 * / 2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 x) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 (x + a)) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  repeat rewrite Rmult_assoc.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 x) * / 2 < Rabs (f2 (x + a)) * (2 * / 2)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 x) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 (x + a)) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite <- Rinv_r_sym.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 x) * / 2 < Rabs (f2 (x + a)) * 1
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
2 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 x) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 (x + a)) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite Rmult_1_r.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 x) * / 2 < Rabs (f2 (x + a))
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
2 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 x) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 (x + a)) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold Rdiv in H5; apply H5.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) * / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (x + a - x) < alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
2 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 x) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 (x + a)) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  replace (x + a - x) with a.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) * / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs a < alp_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) * / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
a = x + a - x
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
2 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 x) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 (x + a)) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H7 := Rlt_le_trans _ _ _ H6 (Rmin_r _ _)); assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) * / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
a = x + a - x
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
2 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 x) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 (x + a)) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  ring.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
2 <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 x) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 (x + a)) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  discrR.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 x) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 (x + a)) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rabs_no_R0; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs (f2 (x + a)) <> 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rabs_no_R0; apply H2.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
H5:forall a0 : R,
Rabs (a0 - x) < alp_f2 ->
Rabs (f2 x) / 2 < Rabs (f2 a0)
a:R
H6:Rabs a < Rmin eps_f2 alp_f2
Rabs a < eps_f2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H7 := Rlt_le_trans _ _ _ H6 (Rmin_l _ _)); assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
forall a : R,
Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
a:R
H5:Rabs (a - x) < alp_f2
Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H6 := H4 a H5).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
a:R
H5:Rabs (a - x) < alp_f2
H6:Rabs (f2 a - f2 x) < Rabs (f2 x) / 2
Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite <- (Rabs_Ropp (f2 a - f2 x)) in H6.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
a:R
H5:Rabs (a - x) < alp_f2
H6:Rabs (- (f2 a - f2 x)) < Rabs (f2 x) / 2
Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite Ropp_minus_distr in H6.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
a:R
H5:Rabs (a - x) < alp_f2
H6:Rabs (f2 x - f2 a) < Rabs (f2 x) / 2
Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  assert (H7 := Rle_lt_trans _ _ _ (Rabs_triang_inv _ _) H6).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
a:R
H5:Rabs (a - x) < alp_f2
H6:Rabs (f2 x - f2 a) < Rabs (f2 x) / 2
H7:Rabs (f2 x) - Rabs (f2 a) < Rabs (f2 x) / 2
Rabs (f2 x) / 2 < Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply Rplus_lt_reg_l with (- Rabs (f2 a) + Rabs (f2 x) / 2).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
a:R
H5:Rabs (a - x) < alp_f2
H6:Rabs (f2 x - f2 a) < Rabs (f2 x) / 2
H7:Rabs (f2 x) - Rabs (f2 a) < Rabs (f2 x) / 2
- Rabs (f2 a) + Rabs (f2 x) / 2 + Rabs (f2 x) / 2 <
- Rabs (f2 a) + Rabs (f2 x) / 2 + Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite Rplus_assoc.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
a:R
H5:Rabs (a - x) < alp_f2
H6:Rabs (f2 x - f2 a) < Rabs (f2 x) / 2
H7:Rabs (f2 x) - Rabs (f2 a) < Rabs (f2 x) / 2
- Rabs (f2 a) + (Rabs (f2 x) / 2 + Rabs (f2 x) / 2) <
- Rabs (f2 a) + Rabs (f2 x) / 2 + Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite <- double_var.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
a:R
H5:Rabs (a - x) < alp_f2
H6:Rabs (f2 x - f2 a) < Rabs (f2 x) / 2
H7:Rabs (f2 x) - Rabs (f2 a) < Rabs (f2 x) / 2
- Rabs (f2 a) + Rabs (f2 x) <
- Rabs (f2 a) + Rabs (f2 x) / 2 + Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  do 2 rewrite (Rplus_comm (- Rabs (f2 a))).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
a:R
H5:Rabs (a - x) < alp_f2
H6:Rabs (f2 x - f2 a) < Rabs (f2 x) / 2
H7:Rabs (f2 x) - Rabs (f2 a) < Rabs (f2 x) / 2
Rabs (f2 x) + - Rabs (f2 a) <
Rabs (f2 x) / 2 + - Rabs (f2 a) + Rabs (f2 a)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
H4:forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
a:R
H5:Rabs (a - x) < alp_f2
H6:Rabs (f2 x - f2 a) < Rabs (f2 x) / 2
H7:Rabs (f2 x) - Rabs (f2 a) < Rabs (f2 x) / 2
Rabs (f2 x) + - Rabs (f2 a) < Rabs (f2 x) / 2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  unfold Rminus in H7; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2 ->
 Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2)
forall x0 : R,
Rabs (x0 - x) < alp_f2 ->
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x1 : R,
 D_x no_cond x x1 /\ Rabs (x1 - x) < alp_f2 ->
 Rabs (f2 x1 - f2 x) < Rabs (f2 x) / 2)
x0:R
H4:Rabs (x0 - x) < alp_f2
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  case (Req_dec x x0); intro.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x1 : R,
 D_x no_cond x x1 /\ Rabs (x1 - x) < alp_f2 ->
 Rabs (f2 x1 - f2 x) < Rabs (f2 x) / 2)
x0:R
H4:Rabs (x0 - x) < alp_f2
H5:x = x0
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x1 : R,
 D_x no_cond x x1 /\ Rabs (x1 - x) < alp_f2 ->
 Rabs (f2 x1 - f2 x) < Rabs (f2 x) / 2)
x0:R
H4:Rabs (x0 - x) < alp_f2
H5:x <> x0
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  rewrite <- H5; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0;
    unfold Rdiv; apply Rmult_lt_0_compat;
      [ apply Rabs_pos_lt; assumption | apply Rinv_0_lt_compat; prove_sup0 ].f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x1 : R,
 D_x no_cond x x1 /\ Rabs (x1 - x) < alp_f2 ->
 Rabs (f2 x1 - f2 x) < Rabs (f2 x) / 2)
x0:R
H4:Rabs (x0 - x) < alp_f2
H5:x <> x0
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  elim H3; intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x1 : R,
 D_x no_cond x x1 /\ Rabs (x1 - x) < alp_f2 ->
 Rabs (f2 x1 - f2 x) < Rabs (f2 x) / 2)
x0:R
H4:Rabs (x0 - x) < alp_f2
H5:x <> x0
H6:alp_f2 > 0
H7:forall x1 : R,
D_x no_cond x x1 /\ Rabs (x1 - x) < alp_f2 ->
Rabs (f2 x1 - f2 x) < Rabs (f2 x) / 2
Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2
  apply H7.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x1 : R,
 D_x no_cond x x1 /\ Rabs (x1 - x) < alp_f2 ->
 Rabs (f2 x1 - f2 x) < Rabs (f2 x) / 2)
x0:R
H4:Rabs (x0 - x) < alp_f2
H5:x <> x0
H6:alp_f2 > 0
H7:forall x1 : R,
D_x no_cond x x1 /\ Rabs (x1 - x) < alp_f2 ->
Rabs (f2 x1 - f2 x) < Rabs (f2 x) / 2
D_x no_cond x x0 /\ Rabs (x0 - x) < alp_f2
  split.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x1 : R,
 D_x no_cond x x1 /\ Rabs (x1 - x) < alp_f2 ->
 Rabs (f2 x1 - f2 x) < Rabs (f2 x) / 2)
x0:R
H4:Rabs (x0 - x) < alp_f2
H5:x <> x0
H6:alp_f2 > 0
H7:forall x1 : R,
D_x no_cond x x1 /\ Rabs (x1 - x) < alp_f2 ->
Rabs (f2 x1 - f2 x) < Rabs (f2 x) / 2
D_x no_cond x x0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x1 : R,
 D_x no_cond x x1 /\ Rabs (x1 - x) < alp_f2 ->
 Rabs (f2 x1 - f2 x) < Rabs (f2 x) / 2)
x0:R
H4:Rabs (x0 - x) < alp_f2
H5:x <> x0
H6:alp_f2 > 0
H7:forall x1 : R,
D_x no_cond x x1 /\ Rabs (x1 - x) < alp_f2 ->
Rabs (f2 x1 - f2 x) < Rabs (f2 x) / 2
Rabs (x0 - x) < alp_f2
  unfold D_x, no_cond; split.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x1 : R,
 D_x no_cond x x1 /\ Rabs (x1 - x) < alp_f2 ->
 Rabs (f2 x1 - f2 x) < Rabs (f2 x) / 2)
x0:R
H4:Rabs (x0 - x) < alp_f2
H5:x <> x0
H6:alp_f2 > 0
H7:forall x1 : R,
D_x no_cond x x1 /\ Rabs (x1 - x) < alp_f2 ->
Rabs (f2 x1 - f2 x) < Rabs (f2 x) / 2
True
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x1 : R,
 D_x no_cond x x1 /\ Rabs (x1 - x) < alp_f2 ->
 Rabs (f2 x1 - f2 x) < Rabs (f2 x) / 2)
x0:R
H4:Rabs (x0 - x) < alp_f2
H5:x <> x0
H6:alp_f2 > 0
H7:forall x1 : R,
D_x no_cond x x1 /\ Rabs (x1 - x) < alp_f2 ->
Rabs (f2 x1 - f2 x) < Rabs (f2 x) / 2
x <> x0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x1 : R,
 D_x no_cond x x1 /\ Rabs (x1 - x) < alp_f2 ->
 Rabs (f2 x1 - f2 x) < Rabs (f2 x) / 2)
x0:R
H4:Rabs (x0 - x) < alp_f2
H5:x <> x0
H6:alp_f2 > 0
H7:forall x1 : R,
D_x no_cond x x1 /\ Rabs (x1 - x) < alp_f2 ->
Rabs (f2 x1 - f2 x) < Rabs (f2 x) / 2
Rabs (x0 - x) < alp_f2
  trivial.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x1 : R,
 D_x no_cond x x1 /\ Rabs (x1 - x) < alp_f2 ->
 Rabs (f2 x1 - f2 x) < Rabs (f2 x) / 2)
x0:R
H4:Rabs (x0 - x) < alp_f2
H5:x <> x0
H6:alp_f2 > 0
H7:forall x1 : R,
D_x no_cond x x1 /\ Rabs (x1 - x) < alp_f2 ->
Rabs (f2 x1 - f2 x) < Rabs (f2 x) / 2
x <> x0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x1 : R,
 D_x no_cond x x1 /\ Rabs (x1 - x) < alp_f2 ->
 Rabs (f2 x1 - f2 x) < Rabs (f2 x) / 2)
x0:R
H4:Rabs (x0 - x) < alp_f2
H5:x <> x0
H6:alp_f2 > 0
H7:forall x1 : R,
D_x no_cond x x1 /\ Rabs (x1 - x) < alp_f2 ->
Rabs (f2 x1 - f2 x) < Rabs (f2 x) / 2
Rabs (x0 - x) < alp_f2
  assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:f2 x <> 0
X:derivable_pt f2 x
eps_f2:posreal
H2:forall h : R, Rabs h < eps_f2 -> f2 (x + h) <> 0
alp_f2:R
H3:alp_f2 > 0 /\
(forall x1 : R,
 D_x no_cond x x1 /\ Rabs (x1 - x) < alp_f2 ->
 Rabs (f2 x1 - f2 x) < Rabs (f2 x) / 2)
x0:R
H4:Rabs (x0 - x) < alp_f2
H5:x <> x0
H6:alp_f2 > 0
H7:forall x1 : R,
D_x no_cond x x1 /\ Rabs (x1 - x) < alp_f2 ->
Rabs (f2 x1 - f2 x) < Rabs (f2 x) / 2
Rabs (x0 - x) < alp_f2
  assumption.
Qed.

Lemma derivable_pt_div :
  forall (f1 f2:R -> R) (x:R),
    derivable_pt f1 x ->
    derivable_pt f2 x -> f2 x <> 0 -> derivable_pt (f1 / f2) x.forall (f1 f2 : R -> R) (x : R),
derivable_pt f1 x ->
derivable_pt f2 x ->
f2 x <> 0 -> derivable_pt (f1 / f2) x
Proof.forall (f1 f2 : R -> R) (x : R),
derivable_pt f1 x ->
derivable_pt f2 x ->
f2 x <> 0 -> derivable_pt (f1 / f2) x
  unfold derivable_pt.forall (f1 f2 : R -> R) (x : R),
{l : R | derivable_pt_abs f1 x l} ->
{l : R | derivable_pt_abs f2 x l} ->
f2 x <> 0 -> {l : R | derivable_pt_abs (f1 / f2) x l}
  intros f1 f2 x X X0 H.f1, f2:R -> R
x:R
X:{l : R | derivable_pt_abs f1 x l}
X0:{l : R | derivable_pt_abs f2 x l}
H:f2 x <> 0
{l : R | derivable_pt_abs (f1 / f2) x l}
  elim X; intros.f1, f2:R -> R
x:R
X:{l : R | derivable_pt_abs f1 x l}
X0:{l : R | derivable_pt_abs f2 x l}
H:f2 x <> 0
x0:R
p:derivable_pt_abs f1 x x0
{l : R | derivable_pt_abs (f1 / f2) x l}
  elim X0; intros.f1, f2:R -> R
x:R
X:{l : R | derivable_pt_abs f1 x l}
X0:{l : R | derivable_pt_abs f2 x l}
H:f2 x <> 0
x0:R
p:derivable_pt_abs f1 x x0
x1:R
p0:derivable_pt_abs f2 x x1
{l : R | derivable_pt_abs (f1 / f2) x l}
  exists ((x0 * f2 x - x1 * f1 x) / Rsqr (f2 x)).f1, f2:R -> R
x:R
X:{l : R | derivable_pt_abs f1 x l}
X0:{l : R | derivable_pt_abs f2 x l}
H:f2 x <> 0
x0:R
p:derivable_pt_abs f1 x x0
x1:R
p0:derivable_pt_abs f2 x x1
derivable_pt_abs (f1 / f2) x
  ((x0 * f2 x - x1 * f1 x) / (f2 x)²)
  apply derivable_pt_lim_div; assumption.
Qed.

Lemma derivable_div :
  forall f1 f2:R -> R,
    derivable f1 ->
    derivable f2 -> (forall x:R, f2 x <> 0) -> derivable (f1 / f2).forall f1 f2 : R -> R,
derivable f1 ->
derivable f2 ->
(forall x : R, f2 x <> 0) -> derivable (f1 / f2)
Proof.forall f1 f2 : R -> R,
derivable f1 ->
derivable f2 ->
(forall x : R, f2 x <> 0) -> derivable (f1 / f2)
  unfold derivable; intros f1 f2 X X0 H x.f1, f2:R -> R
X:forall x0 : R, derivable_pt f1 x0
X0:forall x0 : R, derivable_pt f2 x0
H:forall x0 : R, f2 x0 <> 0
x:R
derivable_pt (f1 / f2) x
  apply (derivable_pt_div _ _ _ (X x) (X0 x) (H x)).
Qed.

Lemma derive_pt_div :
  forall (f1 f2:R -> R) (x:R) (pr1:derivable_pt f1 x)
    (pr2:derivable_pt f2 x) (na:f2 x <> 0),
    derive_pt (f1 / f2) x (derivable_pt_div _ _ _ pr1 pr2 na) =
    (derive_pt f1 x pr1 * f2 x - derive_pt f2 x pr2 * f1 x) / Rsqr (f2 x).forall (f1 f2 : R -> R) (x : R)
  (pr1 : derivable_pt f1 x) (pr2 : derivable_pt f2 x)
  (na : f2 x <> 0),
derive_pt (f1 / f2) x
  (derivable_pt_div f1 f2 x pr1 pr2 na) =
(derive_pt f1 x pr1 * f2 x - derive_pt f2 x pr2 * f1 x) /
(f2 x)²
Proof.forall (f1 f2 : R -> R) (x : R)
  (pr1 : derivable_pt f1 x) (pr2 : derivable_pt f2 x)
  (na : f2 x <> 0),
derive_pt (f1 / f2) x
  (derivable_pt_div f1 f2 x pr1 pr2 na) =
(derive_pt f1 x pr1 * f2 x - derive_pt f2 x pr2 * f1 x) /
(f2 x)²
  intros.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
na:f2 x <> 0
derive_pt (f1 / f2) x
  (derivable_pt_div f1 f2 x pr1 pr2 na) =
(derive_pt f1 x pr1 * f2 x - derive_pt f2 x pr2 * f1 x) /
(f2 x)²
  assert (H := derivable_derive f1 x pr1).f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
na:f2 x <> 0
H:exists l : R, derive_pt f1 x pr1 = l
derive_pt (f1 / f2) x
  (derivable_pt_div f1 f2 x pr1 pr2 na) =
(derive_pt f1 x pr1 * f2 x - derive_pt f2 x pr2 * f1 x) /
(f2 x)²
  assert (H0 := derivable_derive f2 x pr2).f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
na:f2 x <> 0
H:exists l : R, derive_pt f1 x pr1 = l
H0:exists l : R, derive_pt f2 x pr2 = l
derive_pt (f1 / f2) x
  (derivable_pt_div f1 f2 x pr1 pr2 na) =
(derive_pt f1 x pr1 * f2 x - derive_pt f2 x pr2 * f1 x) /
(f2 x)²
  assert
    (H1 := derivable_derive (f1 / f2)%F x (derivable_pt_div _ _ _ pr1 pr2 na)).f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
na:f2 x <> 0
H:exists l : R, derive_pt f1 x pr1 = l
H0:exists l : R, derive_pt f2 x pr2 = l
H1:exists l : R,
  derive_pt (f1 / f2) x
    (derivable_pt_div f1 f2 x pr1 pr2 na) = l
derive_pt (f1 / f2) x
  (derivable_pt_div f1 f2 x pr1 pr2 na) =
(derive_pt f1 x pr1 * f2 x - derive_pt f2 x pr2 * f1 x) /
(f2 x)²
  elim H; clear H; intros l1 H.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
na:f2 x <> 0
H0:exists l : R, derive_pt f2 x pr2 = l
H1:exists l : R,
  derive_pt (f1 / f2) x
    (derivable_pt_div f1 f2 x pr1 pr2 na) = l
l1:R
H:derive_pt f1 x pr1 = l1
derive_pt (f1 / f2) x
  (derivable_pt_div f1 f2 x pr1 pr2 na) =
(derive_pt f1 x pr1 * f2 x - derive_pt f2 x pr2 * f1 x) /
(f2 x)²
  elim H0; clear H0; intros l2 H0.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
na:f2 x <> 0
H1:exists l : R,
  derive_pt (f1 / f2) x
    (derivable_pt_div f1 f2 x pr1 pr2 na) = l
l1:R
H:derive_pt f1 x pr1 = l1
l2:R
H0:derive_pt f2 x pr2 = l2
derive_pt (f1 / f2) x
  (derivable_pt_div f1 f2 x pr1 pr2 na) =
(derive_pt f1 x pr1 * f2 x - derive_pt f2 x pr2 * f1 x) /
(f2 x)²
  elim H1; clear H1; intros l H1.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
na:f2 x <> 0
l1:R
H:derive_pt f1 x pr1 = l1
l2:R
H0:derive_pt f2 x pr2 = l2
l:R
H1:derive_pt (f1 / f2) x
  (derivable_pt_div f1 f2 x pr1 pr2 na) = l
derive_pt (f1 / f2) x
  (derivable_pt_div f1 f2 x pr1 pr2 na) =
(derive_pt f1 x pr1 * f2 x - derive_pt f2 x pr2 * f1 x) /
(f2 x)²
  rewrite H; rewrite H0; apply derive_pt_eq_0.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
na:f2 x <> 0
l1:R
H:derive_pt f1 x pr1 = l1
l2:R
H0:derive_pt f2 x pr2 = l2
l:R
H1:derive_pt (f1 / f2) x
  (derivable_pt_div f1 f2 x pr1 pr2 na) = l
derivable_pt_lim (f1 / f2) x
  ((l1 * f2 x - l2 * f1 x) / (f2 x)²)
  assert (H3 := proj2_sig pr1).f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
na:f2 x <> 0
l1:R
H:derive_pt f1 x pr1 = l1
l2:R
H0:derive_pt f2 x pr2 = l2
l:R
H1:derive_pt (f1 / f2) x
  (derivable_pt_div f1 f2 x pr1 pr2 na) = l
H3:(fun l0 : R => derivable_pt_abs f1 x l0)
  (proj1_sig pr1)
derivable_pt_lim (f1 / f2) x
  ((l1 * f2 x - l2 * f1 x) / (f2 x)²)
  unfold derive_pt in H; rewrite H in H3.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
na:f2 x <> 0
l1:R
H:proj1_sig pr1 = l1
l2:R
H0:derive_pt f2 x pr2 = l2
l:R
H1:derive_pt (f1 / f2) x
  (derivable_pt_div f1 f2 x pr1 pr2 na) = l
H3:derivable_pt_abs f1 x l1
derivable_pt_lim (f1 / f2) x
  ((l1 * f2 x - l2 * f1 x) / (f2 x)²)
  assert (H4 := proj2_sig pr2).f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
na:f2 x <> 0
l1:R
H:proj1_sig pr1 = l1
l2:R
H0:derive_pt f2 x pr2 = l2
l:R
H1:derive_pt (f1 / f2) x
  (derivable_pt_div f1 f2 x pr1 pr2 na) = l
H3:derivable_pt_abs f1 x l1
H4:(fun l0 : R => derivable_pt_abs f2 x l0)
  (proj1_sig pr2)
derivable_pt_lim (f1 / f2) x
  ((l1 * f2 x - l2 * f1 x) / (f2 x)²)
  unfold derive_pt in H0; rewrite H0 in H4.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
na:f2 x <> 0
l1:R
H:proj1_sig pr1 = l1
l2:R
H0:proj1_sig pr2 = l2
l:R
H1:derive_pt (f1 / f2) x
  (derivable_pt_div f1 f2 x pr1 pr2 na) = l
H3:derivable_pt_abs f1 x l1
H4:derivable_pt_abs f2 x l2
derivable_pt_lim (f1 / f2) x
  ((l1 * f2 x - l2 * f1 x) / (f2 x)²)
  apply derivable_pt_lim_div; assumption.
Qed.