Ranalysis2.v

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

(**********)
Lemma formule :
  forall (x h l1 l2:R) (f1 f2:R -> R),
    h <> 0 ->
    f2 x <> 0 ->
    f2 (x + h) <> 0 ->
    (f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
    (l1 * f2 x - l2 * f1 x) / Rsqr (f2 x) =
    / 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 / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x).forall (x h l1 l2 : R) (f1 f2 : R -> R),
h <> 0 ->
f2 x <> 0 ->
f2 (x + h) <> 0 ->
(f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
(l1 * f2 x - l2 * f1 x) / (f2 x)² =
/ 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)
Proof.forall (x h l1 l2 : R) (f1 f2 : R -> R),
h <> 0 ->
f2 x <> 0 ->
f2 (x + h) <> 0 ->
(f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
(l1 * f2 x - l2 * f1 x) / (f2 x)² =
/ 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)
  intros; unfold Rdiv, Rminus, Rsqr.x, h, l1, l2:R
f1, f2:R -> R
H:h <> 0
H0:f2 x <> 0
H1:f2 (x + h) <> 0
(f1 (x + h) * / f2 (x + h) + - (f1 x * / f2 x)) * / h +
- ((l1 * f2 x + - (l2 * f1 x)) * / (f2 x * f2 x)) =
/ 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 * f2 x * f2 (x + h)) *
(f2 (x + h) + - f2 x)
  repeat rewrite Rmult_plus_distr_r; repeat rewrite Rmult_plus_distr_l;
    repeat rewrite Rinv_mult_distr; try assumption.x, h, l1, l2:R
f1, f2:R -> R
H:h <> 0
H0:f2 x <> 0
H1:f2 (x + h) <> 0
f1 (x + h) * / f2 (x + h) * / h +
- (f1 x * / f2 x) * / h +
-
(l1 * f2 x * (/ f2 x * / f2 x) +
 - (l2 * f1 x) * (/ f2 x * / f2 x)) =
/ f2 (x + h) * (f1 (x + h) * / h) +
/ f2 (x + h) * (- f1 x * / h) + / f2 (x + h) * - l1 +
(l1 * (/ f2 x * / f2 (x + h)) * f2 x +
 l1 * (/ f2 x * / f2 (x + h)) * - f2 (x + h)) +
-
(f1 x * (/ f2 x * / f2 (x + h)) * (f2 (x + h) * / h) +
 f1 x * (/ f2 x * / f2 (x + h)) * (- f2 x * / h) +
 f1 x * (/ f2 x * / f2 (x + h)) * - l2) +
(l2 * f1 x * (/ f2 x * / f2 x * / f2 (x + h)) *
 f2 (x + h) +
 l2 * f1 x * (/ f2 x * / f2 x * / f2 (x + h)) * - f2 x)
x, h, l1, l2:R
f1, f2:R -> R
H:h <> 0
H0:f2 x <> 0
H1:f2 (x + h) <> 0
f2 x * f2 x <> 0
  replace (l1 * f2 x * (/ f2 x * / f2 x)) with (l1 * / f2 x * (f2 x * / f2 x));
  [ idtac | ring ].x, h, l1, l2:R
f1, f2:R -> R
H:h <> 0
H0:f2 x <> 0
H1:f2 (x + h) <> 0
f1 (x + h) * / f2 (x + h) * / h +
- (f1 x * / f2 x) * / h +
-
(l1 * / f2 x * (f2 x * / f2 x) +
 - (l2 * f1 x) * (/ f2 x * / f2 x)) =
/ f2 (x + h) * (f1 (x + h) * / h) +
/ f2 (x + h) * (- f1 x * / h) + / f2 (x + h) * - l1 +
(l1 * (/ f2 x * / f2 (x + h)) * f2 x +
 l1 * (/ f2 x * / f2 (x + h)) * - f2 (x + h)) +
-
(f1 x * (/ f2 x * / f2 (x + h)) * (f2 (x + h) * / h) +
 f1 x * (/ f2 x * / f2 (x + h)) * (- f2 x * / h) +
 f1 x * (/ f2 x * / f2 (x + h)) * - l2) +
(l2 * f1 x * (/ f2 x * / f2 x * / f2 (x + h)) *
 f2 (x + h) +
 l2 * f1 x * (/ f2 x * / f2 x * / f2 (x + h)) * - f2 x)
x, h, l1, l2:R
f1, f2:R -> R
H:h <> 0
H0:f2 x <> 0
H1:f2 (x + h) <> 0
f2 x * f2 x <> 0
  replace (l1 * (/ f2 x * / f2 (x + h)) * f2 x) with
  (l1 * / f2 (x + h) * (f2 x * / f2 x)); [ idtac | ring ].x, h, l1, l2:R
f1, f2:R -> R
H:h <> 0
H0:f2 x <> 0
H1:f2 (x + h) <> 0
f1 (x + h) * / f2 (x + h) * / h +
- (f1 x * / f2 x) * / h +
-
(l1 * / f2 x * (f2 x * / f2 x) +
 - (l2 * f1 x) * (/ f2 x * / f2 x)) =
/ f2 (x + h) * (f1 (x + h) * / h) +
/ f2 (x + h) * (- f1 x * / h) + / f2 (x + h) * - l1 +
(l1 * / f2 (x + h) * (f2 x * / f2 x) +
 l1 * (/ f2 x * / f2 (x + h)) * - f2 (x + h)) +
-
(f1 x * (/ f2 x * / f2 (x + h)) * (f2 (x + h) * / h) +
 f1 x * (/ f2 x * / f2 (x + h)) * (- f2 x * / h) +
 f1 x * (/ f2 x * / f2 (x + h)) * - l2) +
(l2 * f1 x * (/ f2 x * / f2 x * / f2 (x + h)) *
 f2 (x + h) +
 l2 * f1 x * (/ f2 x * / f2 x * / f2 (x + h)) * - f2 x)
x, h, l1, l2:R
f1, f2:R -> R
H:h <> 0
H0:f2 x <> 0
H1:f2 (x + h) <> 0
f2 x * f2 x <> 0
  replace (l1 * (/ f2 x * / f2 (x + h)) * - f2 (x + h)) with
  (- (l1 * / f2 x * (f2 (x + h) * / f2 (x + h)))); [ idtac | ring ].x, h, l1, l2:R
f1, f2:R -> R
H:h <> 0
H0:f2 x <> 0
H1:f2 (x + h) <> 0
f1 (x + h) * / f2 (x + h) * / h +
- (f1 x * / f2 x) * / h +
-
(l1 * / f2 x * (f2 x * / f2 x) +
 - (l2 * f1 x) * (/ f2 x * / f2 x)) =
/ f2 (x + h) * (f1 (x + h) * / h) +
/ f2 (x + h) * (- f1 x * / h) + / f2 (x + h) * - l1 +
(l1 * / f2 (x + h) * (f2 x * / f2 x) +
 - (l1 * / f2 x * (f2 (x + h) * / f2 (x + h)))) +
-
(f1 x * (/ f2 x * / f2 (x + h)) * (f2 (x + h) * / h) +
 f1 x * (/ f2 x * / f2 (x + h)) * (- f2 x * / h) +
 f1 x * (/ f2 x * / f2 (x + h)) * - l2) +
(l2 * f1 x * (/ f2 x * / f2 x * / f2 (x + h)) *
 f2 (x + h) +
 l2 * f1 x * (/ f2 x * / f2 x * / f2 (x + h)) * - f2 x)
x, h, l1, l2:R
f1, f2:R -> R
H:h <> 0
H0:f2 x <> 0
H1:f2 (x + h) <> 0
f2 x * f2 x <> 0
  replace (f1 x * (/ f2 x * / f2 (x + h)) * (f2 (x + h) * / h)) with
  (f1 x * / f2 x * / h * (f2 (x + h) * / f2 (x + h)));
  [ idtac | ring ].x, h, l1, l2:R
f1, f2:R -> R
H:h <> 0
H0:f2 x <> 0
H1:f2 (x + h) <> 0
f1 (x + h) * / f2 (x + h) * / h +
- (f1 x * / f2 x) * / h +
-
(l1 * / f2 x * (f2 x * / f2 x) +
 - (l2 * f1 x) * (/ f2 x * / f2 x)) =
/ f2 (x + h) * (f1 (x + h) * / h) +
/ f2 (x + h) * (- f1 x * / h) + / f2 (x + h) * - l1 +
(l1 * / f2 (x + h) * (f2 x * / f2 x) +
 - (l1 * / f2 x * (f2 (x + h) * / f2 (x + h)))) +
-
(f1 x * / f2 x * / h * (f2 (x + h) * / f2 (x + h)) +
 f1 x * (/ f2 x * / f2 (x + h)) * (- f2 x * / h) +
 f1 x * (/ f2 x * / f2 (x + h)) * - l2) +
(l2 * f1 x * (/ f2 x * / f2 x * / f2 (x + h)) *
 f2 (x + h) +
 l2 * f1 x * (/ f2 x * / f2 x * / f2 (x + h)) * - f2 x)
x, h, l1, l2:R
f1, f2:R -> R
H:h <> 0
H0:f2 x <> 0
H1:f2 (x + h) <> 0
f2 x * f2 x <> 0
  replace (f1 x * (/ f2 x * / f2 (x + h)) * (- f2 x * / h)) with
  (- (f1 x * / f2 (x + h) * / h * (f2 x * / f2 x)));
  [ idtac | ring ].x, h, l1, l2:R
f1, f2:R -> R
H:h <> 0
H0:f2 x <> 0
H1:f2 (x + h) <> 0
f1 (x + h) * / f2 (x + h) * / h +
- (f1 x * / f2 x) * / h +
-
(l1 * / f2 x * (f2 x * / f2 x) +
 - (l2 * f1 x) * (/ f2 x * / f2 x)) =
/ f2 (x + h) * (f1 (x + h) * / h) +
/ f2 (x + h) * (- f1 x * / h) + / f2 (x + h) * - l1 +
(l1 * / f2 (x + h) * (f2 x * / f2 x) +
 - (l1 * / f2 x * (f2 (x + h) * / f2 (x + h)))) +
-
(f1 x * / f2 x * / h * (f2 (x + h) * / f2 (x + h)) +
 - (f1 x * / f2 (x + h) * / h * (f2 x * / f2 x)) +
 f1 x * (/ f2 x * / f2 (x + h)) * - l2) +
(l2 * f1 x * (/ f2 x * / f2 x * / f2 (x + h)) *
 f2 (x + h) +
 l2 * f1 x * (/ f2 x * / f2 x * / f2 (x + h)) * - f2 x)
x, h, l1, l2:R
f1, f2:R -> R
H:h <> 0
H0:f2 x <> 0
H1:f2 (x + h) <> 0
f2 x * f2 x <> 0
  replace (l2 * f1 x * (/ f2 x * / f2 x * / f2 (x + h)) * f2 (x + h)) with
  (l2 * f1 x * / f2 x * / f2 x * (f2 (x + h) * / f2 (x + h)));
  [ idtac | ring ].x, h, l1, l2:R
f1, f2:R -> R
H:h <> 0
H0:f2 x <> 0
H1:f2 (x + h) <> 0
f1 (x + h) * / f2 (x + h) * / h +
- (f1 x * / f2 x) * / h +
-
(l1 * / f2 x * (f2 x * / f2 x) +
 - (l2 * f1 x) * (/ f2 x * / f2 x)) =
/ f2 (x + h) * (f1 (x + h) * / h) +
/ f2 (x + h) * (- f1 x * / h) + / f2 (x + h) * - l1 +
(l1 * / f2 (x + h) * (f2 x * / f2 x) +
 - (l1 * / f2 x * (f2 (x + h) * / f2 (x + h)))) +
-
(f1 x * / f2 x * / h * (f2 (x + h) * / f2 (x + h)) +
 - (f1 x * / f2 (x + h) * / h * (f2 x * / f2 x)) +
 f1 x * (/ f2 x * / f2 (x + h)) * - l2) +
(l2 * f1 x * / f2 x * / f2 x *
 (f2 (x + h) * / f2 (x + h)) +
 l2 * f1 x * (/ f2 x * / f2 x * / f2 (x + h)) * - f2 x)
x, h, l1, l2:R
f1, f2:R -> R
H:h <> 0
H0:f2 x <> 0
H1:f2 (x + h) <> 0
f2 x * f2 x <> 0
  replace (l2 * f1 x * (/ f2 x * / f2 x * / f2 (x + h)) * - f2 x) with
  (- (l2 * f1 x * / f2 x * / f2 (x + h) * (f2 x * / f2 x)));
  [ idtac | ring ].x, h, l1, l2:R
f1, f2:R -> R
H:h <> 0
H0:f2 x <> 0
H1:f2 (x + h) <> 0
f1 (x + h) * / f2 (x + h) * / h +
- (f1 x * / f2 x) * / h +
-
(l1 * / f2 x * (f2 x * / f2 x) +
 - (l2 * f1 x) * (/ f2 x * / f2 x)) =
/ f2 (x + h) * (f1 (x + h) * / h) +
/ f2 (x + h) * (- f1 x * / h) + / f2 (x + h) * - l1 +
(l1 * / f2 (x + h) * (f2 x * / f2 x) +
 - (l1 * / f2 x * (f2 (x + h) * / f2 (x + h)))) +
-
(f1 x * / f2 x * / h * (f2 (x + h) * / f2 (x + h)) +
 - (f1 x * / f2 (x + h) * / h * (f2 x * / f2 x)) +
 f1 x * (/ f2 x * / f2 (x + h)) * - l2) +
(l2 * f1 x * / f2 x * / f2 x *
 (f2 (x + h) * / f2 (x + h)) +
 -
 (l2 * f1 x * / f2 x * / f2 (x + h) * (f2 x * / f2 x)))
x, h, l1, l2:R
f1, f2:R -> R
H:h <> 0
H0:f2 x <> 0
H1:f2 (x + h) <> 0
f2 x * f2 x <> 0
  repeat rewrite <- Rinv_r_sym; try assumption || ring.x, h, l1, l2:R
f1, f2:R -> R
H:h <> 0
H0:f2 x <> 0
H1:f2 (x + h) <> 0
f2 x * f2 x <> 0
  apply prod_neq_R0; assumption.
Qed.

(* begin hide *)
Notation Rmin_pos := Rmin_pos (only parsing). (* compat *)
(* end hide *)

Lemma maj_term1 :
  forall (x h eps l1 alp_f2:R) (eps_f2 alp_f1d:posreal)
    (f1 f2:R -> R),
    0 < eps ->
    f2 x <> 0 ->
    f2 (x + h) <> 0 ->
    (forall h:R,
      h <> 0 ->
      Rabs h < alp_f1d ->
      Rabs ((f1 (x + h) - f1 x) / h - l1) < Rabs (eps * f2 x / 8)) ->
    (forall a:R,
      Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) ->
    h <> 0 ->
    Rabs h < alp_f1d ->
    Rabs h < Rmin eps_f2 alp_f2 ->
    Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) < eps / 4.forall (x h eps l1 alp_f2 : R)
  (eps_f2 alp_f1d : posreal) (f1 f2 : R -> R),
0 < eps ->
f2 x <> 0 ->
f2 (x + h) <> 0 ->
(forall h0 : R,
 h0 <> 0 ->
 Rabs h0 < alp_f1d ->
 Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
 Rabs (eps * f2 x / 8)) ->
(forall a : R,
 Rabs a < Rmin eps_f2 alp_f2 ->
 / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) ->
h <> 0 ->
Rabs h < alp_f1d ->
Rabs h < Rmin eps_f2 alp_f2 ->
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) <
eps / 4
Proof.forall (x h eps l1 alp_f2 : R)
  (eps_f2 alp_f1d : posreal) (f1 f2 : R -> R),
0 < eps ->
f2 x <> 0 ->
f2 (x + h) <> 0 ->
(forall h0 : R,
 h0 <> 0 ->
 Rabs h0 < alp_f1d ->
 Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
 Rabs (eps * f2 x / 8)) ->
(forall a : R,
 Rabs a < Rmin eps_f2 alp_f2 ->
 / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) ->
h <> 0 ->
Rabs h < alp_f1d ->
Rabs h < Rmin eps_f2 alp_f2 ->
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) <
eps / 4
  intros.x, h, eps, l1, alp_f2:R
eps_f2, alp_f1d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f1d
H6:Rabs h < Rmin eps_f2 alp_f2
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) <
eps / 4
  assert (H7 := H3 h H6).x, h, eps, l1, alp_f2:R
eps_f2, alp_f1d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f1d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) <
eps / 4
  assert (H8 := H2 h H4 H5).x, h, eps, l1, alp_f2:R
eps_f2, alp_f1d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f1d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H8: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
  apply Rle_lt_trans with
    (2 / Rabs (f2 x) * Rabs ((f1 (x + h) - f1 x) / h - l1)).x, h, eps, l1, alp_f2:R
eps_f2, alp_f1d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f1d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H8:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) <=
2 / Rabs (f2 x) * Rabs ((f1 (x + h) - f1 x) / h - l1)
x, h, eps, l1, alp_f2:R
eps_f2, alp_f1d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f1d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H8:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
2 / Rabs (f2 x) * Rabs ((f1 (x + h) - f1 x) / h - l1) <
eps / 4
  rewrite Rabs_mult.x, h, eps, l1, alp_f2:R
eps_f2, alp_f1d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f1d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H8: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) <=
2 / Rabs (f2 x) * Rabs ((f1 (x + h) - f1 x) / h - l1)
x, h, eps, l1, alp_f2:R
eps_f2, alp_f1d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f1d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H8:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
2 / Rabs (f2 x) * Rabs ((f1 (x + h) - f1 x) / h - l1) <
eps / 4
  apply Rmult_le_compat_r.x, h, eps, l1, alp_f2:R
eps_f2, alp_f1d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f1d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H8:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
0 <= Rabs ((f1 (x + h) - f1 x) / h - l1)
x, h, eps, l1, alp_f2:R
eps_f2, alp_f1d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f1d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H8:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (/ f2 (x + h)) <= 2 / Rabs (f2 x)
x, h, eps, l1, alp_f2:R
eps_f2, alp_f1d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f1d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H8:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
2 / Rabs (f2 x) * Rabs ((f1 (x + h) - f1 x) / h - l1) <
eps / 4
  apply Rabs_pos.x, h, eps, l1, alp_f2:R
eps_f2, alp_f1d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f1d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H8:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (/ f2 (x + h)) <= 2 / Rabs (f2 x)
x, h, eps, l1, alp_f2:R
eps_f2, alp_f1d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f1d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H8:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
2 / Rabs (f2 x) * Rabs ((f1 (x + h) - f1 x) / h - l1) <
eps / 4
  rewrite Rabs_Rinv; [ left; exact H7 | assumption ].x, h, eps, l1, alp_f2:R
eps_f2, alp_f1d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f1d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H8:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
2 / Rabs (f2 x) * Rabs ((f1 (x + h) - f1 x) / h - l1) <
eps / 4
  apply Rlt_le_trans with (2 / Rabs (f2 x) * Rabs (eps * f2 x / 8)).x, h, eps, l1, alp_f2:R
eps_f2, alp_f1d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f1d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H8:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
2 / Rabs (f2 x) * Rabs ((f1 (x + h) - f1 x) / h - l1) <
2 / Rabs (f2 x) * Rabs (eps * f2 x / 8)
x, h, eps, l1, alp_f2:R
eps_f2, alp_f1d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f1d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H8:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
2 / Rabs (f2 x) * Rabs (eps * f2 x / 8) <= eps / 4
  apply Rmult_lt_compat_l.x, h, eps, l1, alp_f2:R
eps_f2, alp_f1d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f1d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H8:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
0 < 2 / Rabs (f2 x)
x, h, eps, l1, alp_f2:R
eps_f2, alp_f1d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f1d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H8:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
x, h, eps, l1, alp_f2:R
eps_f2, alp_f1d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f1d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H8:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
2 / Rabs (f2 x) * Rabs (eps * f2 x / 8) <= eps / 4
  unfold Rdiv; apply Rmult_lt_0_compat;
    [ prove_sup0 | apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption ].x, h, eps, l1, alp_f2:R
eps_f2, alp_f1d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f1d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H8:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
x, h, eps, l1, alp_f2:R
eps_f2, alp_f1d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f1d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H8:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
2 / Rabs (f2 x) * Rabs (eps * f2 x / 8) <= eps / 4
  exact H8.x, h, eps, l1, alp_f2:R
eps_f2, alp_f1d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f1d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H8:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
2 / Rabs (f2 x) * Rabs (eps * f2 x / 8) <= eps / 4
  right; unfold Rdiv.x, h, eps, l1, alp_f2:R
eps_f2, alp_f1d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f1d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H8:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
2 * / Rabs (f2 x) * Rabs (eps * f2 x * / 8) =
eps * / 4
  repeat rewrite Rabs_mult.x, h, eps, l1, alp_f2:R
eps_f2, alp_f1d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f1d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H8:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
2 * / Rabs (f2 x) *
(Rabs eps * Rabs (f2 x) * Rabs (/ 8)) = eps * / 4
  rewrite Rabs_Rinv; discrR.x, h, eps, l1, alp_f2:R
eps_f2, alp_f1d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f1d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H8:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
2 * / Rabs (f2 x) *
(Rabs eps * Rabs (f2 x) * / Rabs 8) = eps * / 4
  rewrite (Rabs_pos_eq 8) by now apply IZR_le.x, h, eps, l1, alp_f2:R
eps_f2, alp_f1d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f1d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H8:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
2 * / Rabs (f2 x) * (Rabs eps * Rabs (f2 x) * / 8) =
eps * / 4
  rewrite (Rabs_pos_eq eps).x, h, eps, l1, alp_f2:R
eps_f2, alp_f1d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f1d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H8:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
2 * / Rabs (f2 x) * (eps * Rabs (f2 x) * / 8) =
eps * / 4
x, h, eps, l1, alp_f2:R
eps_f2, alp_f1d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f1d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H8:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
0 <= eps
  field.x, h, eps, l1, alp_f2:R
eps_f2, alp_f1d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f1d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H8:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
Rabs (f2 x) <> 0
x, h, eps, l1, alp_f2:R
eps_f2, alp_f1d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f1d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H8:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
0 <= eps
  now apply Rabs_no_R0.x, h, eps, l1, alp_f2:R
eps_f2, alp_f1d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f1d ->
Rabs ((f1 (x + h0) - f1 x) / h0 - l1) <
Rabs (eps * f2 x / 8)
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f1d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H8:Rabs ((f1 (x + h) - f1 x) / h - l1) <
Rabs (eps * f2 x / 8)
0 <= eps
  now apply Rlt_le.
Qed.

Lemma maj_term2 :
  forall (x h eps l1 alp_f2 alp_f2t2:R) (eps_f2:posreal)
    (f2:R -> R),
    0 < eps ->
    f2 x <> 0 ->
    f2 (x + h) <> 0 ->
    (forall a:R,
      Rabs a < alp_f2t2 ->
      Rabs (f2 (x + a) - f2 x) < Rabs (eps * Rsqr (f2 x) / (8 * l1))) ->
    (forall a:R,
      Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) ->
    h <> 0 ->
    Rabs h < alp_f2t2 ->
    Rabs h < Rmin eps_f2 alp_f2 ->
    l1 <> 0 -> Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) < eps / 4.forall (x h eps l1 alp_f2 alp_f2t2 : R)
  (eps_f2 : posreal) (f2 : R -> R),
0 < eps ->
f2 x <> 0 ->
f2 (x + h) <> 0 ->
(forall a : R,
 Rabs a < alp_f2t2 ->
 Rabs (f2 (x + a) - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1))) ->
(forall a : R,
 Rabs a < Rmin eps_f2 alp_f2 ->
 / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) ->
h <> 0 ->
Rabs h < alp_f2t2 ->
Rabs h < Rmin eps_f2 alp_f2 ->
l1 <> 0 ->
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) <
eps / 4
Proof.forall (x h eps l1 alp_f2 alp_f2t2 : R)
  (eps_f2 : posreal) (f2 : R -> R),
0 < eps ->
f2 x <> 0 ->
f2 (x + h) <> 0 ->
(forall a : R,
 Rabs a < alp_f2t2 ->
 Rabs (f2 (x + a) - f2 x) <
 Rabs (eps * (f2 x)² / (8 * l1))) ->
(forall a : R,
 Rabs a < Rmin eps_f2 alp_f2 ->
 / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) ->
h <> 0 ->
Rabs h < alp_f2t2 ->
Rabs h < Rmin eps_f2 alp_f2 ->
l1 <> 0 ->
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) <
eps / 4
  intros.x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) <
eps / 4
  assert (H8 := H3 h H6).x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) <
eps / 4
  assert (H9 := H2 h H5).x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) <
eps / 4
  apply Rle_lt_trans with
    (Rabs (l1 / (f2 x * f2 (x + h))) * Rabs (eps * Rsqr (f2 x) / (8 * l1))).x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) <=
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (eps * (f2 x)² / (8 * l1))
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (eps * (f2 x)² / (8 * l1)) < 
eps / 4
  rewrite Rabs_mult; apply Rmult_le_compat_l.x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 <= Rabs (l1 / (f2 x * f2 (x + h)))
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (f2 x - f2 (x + h)) <=
Rabs (eps * (f2 x)² / (8 * l1))
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (eps * (f2 x)² / (8 * l1)) < 
eps / 4
  apply Rabs_pos.x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (f2 x - f2 (x + h)) <=
Rabs (eps * (f2 x)² / (8 * l1))
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (eps * (f2 x)² / (8 * l1)) < 
eps / 4
  rewrite <- (Rabs_Ropp (f2 x - f2 (x + h))); rewrite Ropp_minus_distr.x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (f2 (x + h) - f2 x) <=
Rabs (eps * (f2 x)² / (8 * l1))
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (eps * (f2 x)² / (8 * l1)) < 
eps / 4
  left; apply H9.x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (eps * (f2 x)² / (8 * l1)) < eps / 4
  apply Rlt_le_trans with
    (Rabs (2 * (l1 / (f2 x * f2 x))) * Rabs (eps * Rsqr (f2 x) / (8 * l1))).x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l1 / (f2 x * f2 (x + h))) *
Rabs (eps * (f2 x)² / (8 * l1)) <
Rabs (2 * (l1 / (f2 x * f2 x))) *
Rabs (eps * (f2 x)² / (8 * l1))
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (2 * (l1 / (f2 x * f2 x))) *
Rabs (eps * (f2 x)² / (8 * l1)) <= 
eps / 4
  apply Rmult_lt_compat_r.x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rabs (eps * (f2 x)² / (8 * l1))
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l1 / (f2 x * f2 (x + h))) <
Rabs (2 * (l1 / (f2 x * f2 x)))
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (2 * (l1 / (f2 x * f2 x))) *
Rabs (eps * (f2 x)² / (8 * l1)) <= 
eps / 4
  apply Rabs_pos_lt.x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps * (f2 x)² / (8 * l1) <> 0
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l1 / (f2 x * f2 (x + h))) <
Rabs (2 * (l1 / (f2 x * f2 x)))
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (2 * (l1 / (f2 x * f2 x))) *
Rabs (eps * (f2 x)² / (8 * l1)) <= 
eps / 4
  unfold Rdiv; unfold Rsqr; repeat apply prod_neq_R0;
    try assumption || discrR.x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps <> 0
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
/ (8 * l1) <> 0
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l1 / (f2 x * f2 (x + h))) <
Rabs (2 * (l1 / (f2 x * f2 x)))
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (2 * (l1 / (f2 x * f2 x))) *
Rabs (eps * (f2 x)² / (8 * l1)) <= 
eps / 4
  red; intro H10; rewrite H10 in H; elim (Rlt_irrefl _ H).x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
/ (8 * l1) <> 0
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l1 / (f2 x * f2 (x + h))) <
Rabs (2 * (l1 / (f2 x * f2 x)))
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (2 * (l1 / (f2 x * f2 x))) *
Rabs (eps * (f2 x)² / (8 * l1)) <= 
eps / 4
  apply Rinv_neq_0_compat; apply prod_neq_R0; try assumption || discrR.x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l1 / (f2 x * f2 (x + h))) <
Rabs (2 * (l1 / (f2 x * f2 x)))
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (2 * (l1 / (f2 x * f2 x))) *
Rabs (eps * (f2 x)² / (8 * l1)) <= 
eps / 4
  unfold Rdiv.x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l1 * / (f2 x * f2 (x + h))) <
Rabs (2 * (l1 * / (f2 x * f2 x)))
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (2 * (l1 / (f2 x * f2 x))) *
Rabs (eps * (f2 x)² / (8 * l1)) <= 
eps / 4
  repeat rewrite Rinv_mult_distr; try assumption.x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (l1 * (/ f2 x * / f2 (x + h))) <
Rabs (2 * (l1 * (/ f2 x * / f2 x)))
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (2 * (l1 / (f2 x * f2 x))) *
Rabs (eps * (f2 x)² / (8 * l1)) <= 
eps / 4
  repeat rewrite Rabs_mult.x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs l1 * (Rabs (/ f2 x) * Rabs (/ f2 (x + h))) <
Rabs 2 * (Rabs l1 * (Rabs (/ f2 x) * Rabs (/ f2 x)))
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (2 * (l1 / (f2 x * f2 x))) *
Rabs (eps * (f2 x)² / (8 * l1)) <= 
eps / 4
  replace (Rabs 2) with 2.x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs l1 * (Rabs (/ f2 x) * Rabs (/ f2 (x + h))) <
2 * (Rabs l1 * (Rabs (/ f2 x) * Rabs (/ f2 x)))
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
2 = Rabs 2
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (2 * (l1 / (f2 x * f2 x))) *
Rabs (eps * (f2 x)² / (8 * l1)) <= 
eps / 4
  rewrite (Rmult_comm 2).x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs l1 * (Rabs (/ f2 x) * Rabs (/ f2 (x + h))) <
Rabs l1 * (Rabs (/ f2 x) * Rabs (/ f2 x)) * 2
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
2 = Rabs 2
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (2 * (l1 / (f2 x * f2 x))) *
Rabs (eps * (f2 x)² / (8 * l1)) <= 
eps / 4
  replace (Rabs l1 * (Rabs (/ f2 x) * Rabs (/ f2 x)) * 2) with
  (Rabs l1 * (Rabs (/ f2 x) * (Rabs (/ f2 x) * 2)));
  [ idtac | ring ].x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs l1 * (Rabs (/ f2 x) * Rabs (/ f2 (x + h))) <
Rabs l1 * (Rabs (/ f2 x) * (Rabs (/ f2 x) * 2))
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
2 = Rabs 2
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (2 * (l1 / (f2 x * f2 x))) *
Rabs (eps * (f2 x)² / (8 * l1)) <= 
eps / 4
  repeat apply Rmult_lt_compat_l.x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rabs l1
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rabs (/ f2 x)
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (/ f2 (x + h)) < Rabs (/ f2 x) * 2
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
2 = Rabs 2
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (2 * (l1 / (f2 x * f2 x))) *
Rabs (eps * (f2 x)² / (8 * l1)) <= 
eps / 4
  apply Rabs_pos_lt; assumption.x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
0 < Rabs (/ f2 x)
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (/ f2 (x + h)) < Rabs (/ f2 x) * 2
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
2 = Rabs 2
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (2 * (l1 / (f2 x * f2 x))) *
Rabs (eps * (f2 x)² / (8 * l1)) <= 
eps / 4
  apply Rabs_pos_lt; apply Rinv_neq_0_compat; assumption.x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (/ f2 (x + h)) < Rabs (/ f2 x) * 2
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
2 = Rabs 2
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (2 * (l1 / (f2 x * f2 x))) *
Rabs (eps * (f2 x)² / (8 * l1)) <= 
eps / 4
  repeat rewrite Rabs_Rinv; try assumption.x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
/ Rabs (f2 (x + h)) < / Rabs (f2 x) * 2
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
2 = Rabs 2
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (2 * (l1 / (f2 x * f2 x))) *
Rabs (eps * (f2 x)² / (8 * l1)) <= 
eps / 4
  rewrite <- (Rmult_comm 2).x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
/ Rabs (f2 (x + h)) < 2 * / Rabs (f2 x)
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
2 = Rabs 2
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (2 * (l1 / (f2 x * f2 x))) *
Rabs (eps * (f2 x)² / (8 * l1)) <= 
eps / 4
  unfold Rdiv in H8; exact H8.x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
2 = Rabs 2
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (2 * (l1 / (f2 x * f2 x))) *
Rabs (eps * (f2 x)² / (8 * l1)) <= 
eps / 4
  symmetry ; apply Rabs_right; left; prove_sup0.x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (2 * (l1 / (f2 x * f2 x))) *
Rabs (eps * (f2 x)² / (8 * l1)) <= eps / 4
  right.x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (2 * (l1 / (f2 x * f2 x))) *
Rabs (eps * (f2 x)² / (8 * l1)) = eps / 4
  unfold Rsqr, Rdiv.x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (2 * (l1 * / (f2 x * f2 x))) *
Rabs (eps * (f2 x * f2 x) * / (8 * l1)) = eps * / 4
  do 1 rewrite Rinv_mult_distr; try assumption || discrR.x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (2 * (l1 * (/ f2 x * / f2 x))) *
Rabs (eps * (f2 x * f2 x) * / (8 * l1)) = eps * / 4
  do 1 rewrite Rinv_mult_distr; try assumption || discrR.x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs (2 * (l1 * (/ f2 x * / f2 x))) *
Rabs (eps * (f2 x * f2 x) * (/ 8 * / l1)) = eps * / 4
  repeat rewrite Rabs_mult.x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs 2 * (Rabs l1 * (Rabs (/ f2 x) * Rabs (/ f2 x))) *
(Rabs eps * (Rabs (f2 x) * Rabs (f2 x)) *
 (Rabs (/ 8) * Rabs (/ l1))) = eps * / 4
  repeat rewrite Rabs_Rinv; try assumption || discrR.x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs 2 * (Rabs l1 * (/ Rabs (f2 x) * / Rabs (f2 x))) *
(Rabs eps * (Rabs (f2 x) * Rabs (f2 x)) *
 (/ Rabs 8 * / Rabs l1)) = eps * / 4
  replace (Rabs eps) with eps.x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs 2 * (Rabs l1 * (/ Rabs (f2 x) * / Rabs (f2 x))) *
(eps * (Rabs (f2 x) * Rabs (f2 x)) *
 (/ Rabs 8 * / Rabs l1)) = eps * / 4
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps = Rabs eps
  replace (Rabs 8) with 8.x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
Rabs 2 * (Rabs l1 * (/ Rabs (f2 x) * / Rabs (f2 x))) *
(eps * (Rabs (f2 x) * Rabs (f2 x)) * (/ 8 * / Rabs l1)) =
eps * / 4
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
8 = Rabs 8
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps = Rabs eps
  replace (Rabs 2) with 2.x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
2 * (Rabs l1 * (/ Rabs (f2 x) * / Rabs (f2 x))) *
(eps * (Rabs (f2 x) * Rabs (f2 x)) * (/ 8 * / Rabs l1)) =
eps * / 4
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
2 = Rabs 2
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
8 = Rabs 8
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps = Rabs eps
  replace 8 with (4 * 2); [ idtac | ring ].x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
2 * (Rabs l1 * (/ Rabs (f2 x) * / Rabs (f2 x))) *
(eps * (Rabs (f2 x) * Rabs (f2 x)) *
 (/ (4 * 2) * / Rabs l1)) = eps * / 4
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
2 = Rabs 2
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
8 = Rabs 8
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps = Rabs eps
  rewrite Rinv_mult_distr; discrR.x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
2 * (Rabs l1 * (/ Rabs (f2 x) * / Rabs (f2 x))) *
(eps * (Rabs (f2 x) * Rabs (f2 x)) *
 (/ 4 * / 2 * / Rabs l1)) = eps * / 4
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
2 = Rabs 2
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
8 = Rabs 8
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps = Rabs eps
  replace
  (2 * (Rabs l1 * (/ Rabs (f2 x) * / Rabs (f2 x))) *
    (eps * (Rabs (f2 x) * Rabs (f2 x)) * (/ 4 * / 2 * / Rabs l1))) with
  (eps * / 4 * (Rabs l1 * / Rabs l1) * (Rabs (f2 x) * / Rabs (f2 x)) *
    (Rabs (f2 x) * / Rabs (f2 x)) * (2 * / 2)); [ idtac | ring ].x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps * / 4 * (Rabs l1 * / Rabs l1) *
(Rabs (f2 x) * / Rabs (f2 x)) *
(Rabs (f2 x) * / Rabs (f2 x)) * (2 * / 2) = eps * / 4
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
2 = Rabs 2
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
8 = Rabs 8
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps = Rabs eps
  repeat rewrite <- Rinv_r_sym; try (apply Rabs_no_R0; assumption) || discrR.x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps * / 4 * 1 * 1 * 1 * 1 = eps * / 4
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
2 = Rabs 2
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
8 = Rabs 8
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps = Rabs eps
  ring.x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
2 = Rabs 2
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
8 = Rabs 8
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps = Rabs eps
  symmetry ; apply Rabs_right; left; prove_sup0.x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
8 = Rabs 8
x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps = Rabs eps
  symmetry ; apply Rabs_right; left; prove_sup.x, h, eps, l1, alp_f2, alp_f2t2:R
eps_f2:posreal
f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2t2
H6:Rabs h < Rmin eps_f2 alp_f2
H7:l1 <> 0
H8:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9:Rabs (f2 (x + h) - f2 x) <
Rabs (eps * (f2 x)² / (8 * l1))
eps = Rabs eps
  symmetry ; apply Rabs_right; left; assumption.
Qed.

Lemma maj_term3 :
  forall (x h eps l2 alp_f2:R) (eps_f2 alp_f2d:posreal)
    (f1 f2:R -> R),
    0 < eps ->
    f2 x <> 0 ->
    f2 (x + h) <> 0 ->
    (forall h:R,
      h <> 0 ->
      Rabs h < alp_f2d ->
      Rabs ((f2 (x + h) - f2 x) / h - l2) <
      Rabs (Rsqr (f2 x) * eps / (8 * f1 x))) ->
    (forall a:R,
      Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) ->
    h <> 0 ->
    Rabs h < alp_f2d ->
    Rabs h < Rmin eps_f2 alp_f2 ->
    f1 x <> 0 ->
    Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) <
    eps / 4.forall (x h eps l2 alp_f2 : R)
  (eps_f2 alp_f2d : posreal) (f1 f2 : R -> R),
0 < eps ->
f2 x <> 0 ->
f2 (x + h) <> 0 ->
(forall h0 : R,
 h0 <> 0 ->
 Rabs h0 < alp_f2d ->
 Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
 Rabs ((f2 x)² * eps / (8 * f1 x))) ->
(forall a : R,
 Rabs a < Rmin eps_f2 alp_f2 ->
 / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) ->
h <> 0 ->
Rabs h < alp_f2d ->
Rabs h < Rmin eps_f2 alp_f2 ->
f1 x <> 0 ->
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) < eps / 4
Proof.forall (x h eps l2 alp_f2 : R)
  (eps_f2 alp_f2d : posreal) (f1 f2 : R -> R),
0 < eps ->
f2 x <> 0 ->
f2 (x + h) <> 0 ->
(forall h0 : R,
 h0 <> 0 ->
 Rabs h0 < alp_f2d ->
 Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
 Rabs ((f2 x)² * eps / (8 * f1 x))) ->
(forall a : R,
 Rabs a < Rmin eps_f2 alp_f2 ->
 / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) ->
h <> 0 ->
Rabs h < alp_f2d ->
Rabs h < Rmin eps_f2 alp_f2 ->
f1 x <> 0 ->
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) < eps / 4
  intros.x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) < eps / 4
  assert (H8 := H2 h H4 H5).x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) < eps / 4
  assert (H9 := H3 h H6).x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) < eps / 4
  apply Rle_lt_trans with
    (Rabs (f1 x / (f2 x * f2 (x + h))) * Rabs (Rsqr (f2 x) * eps / (8 * f1 x))).x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs
  (f1 x / (f2 x * f2 (x + h)) *
   ((f2 (x + h) - f2 x) / h - l2)) <=
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 x)² * eps / (8 * f1 x))
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 x)² * eps / (8 * f1 x)) < 
eps / 4
  rewrite Rabs_mult.x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 (x + h) - f2 x) / h - l2) <=
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 x)² * eps / (8 * f1 x))
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 x)² * eps / (8 * f1 x)) < 
eps / 4
  apply Rmult_le_compat_l.x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
0 <= Rabs (f1 x / (f2 x * f2 (x + h)))
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs ((f2 (x + h) - f2 x) / h - l2) <=
Rabs ((f2 x)² * eps / (8 * f1 x))
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 x)² * eps / (8 * f1 x)) < 
eps / 4
  apply Rabs_pos.x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs ((f2 (x + h) - f2 x) / h - l2) <=
Rabs ((f2 x)² * eps / (8 * f1 x))
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 x)² * eps / (8 * f1 x)) < 
eps / 4
  left; apply H8.x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 x)² * eps / (8 * f1 x)) < eps / 4
  apply Rlt_le_trans with
    (Rabs (2 * (f1 x / (f2 x * f2 x))) * Rabs (Rsqr (f2 x) * eps / (8 * f1 x))).x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (f1 x / (f2 x * f2 (x + h))) *
Rabs ((f2 x)² * eps / (8 * f1 x)) <
Rabs (2 * (f1 x / (f2 x * f2 x))) *
Rabs ((f2 x)² * eps / (8 * f1 x))
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * (f1 x / (f2 x * f2 x))) *
Rabs ((f2 x)² * eps / (8 * f1 x)) <= 
eps / 4
  apply Rmult_lt_compat_r.x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
0 < Rabs ((f2 x)² * eps / (8 * f1 x))
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (f1 x / (f2 x * f2 (x + h))) <
Rabs (2 * (f1 x / (f2 x * f2 x)))
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * (f1 x / (f2 x * f2 x))) *
Rabs ((f2 x)² * eps / (8 * f1 x)) <= 
eps / 4
  apply Rabs_pos_lt.x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
(f2 x)² * eps / (8 * f1 x) <> 0
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (f1 x / (f2 x * f2 (x + h))) <
Rabs (2 * (f1 x / (f2 x * f2 x)))
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * (f1 x / (f2 x * f2 x))) *
Rabs ((f2 x)² * eps / (8 * f1 x)) <= 
eps / 4
  unfold Rdiv; unfold Rsqr; repeat apply prod_neq_R0;
    try assumption.x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
eps <> 0
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
/ (8 * f1 x) <> 0
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (f1 x / (f2 x * f2 (x + h))) <
Rabs (2 * (f1 x / (f2 x * f2 x)))
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * (f1 x / (f2 x * f2 x))) *
Rabs ((f2 x)² * eps / (8 * f1 x)) <= 
eps / 4
  red; intro H10; rewrite H10 in H; elim (Rlt_irrefl _ H).x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
/ (8 * f1 x) <> 0
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (f1 x / (f2 x * f2 (x + h))) <
Rabs (2 * (f1 x / (f2 x * f2 x)))
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * (f1 x / (f2 x * f2 x))) *
Rabs ((f2 x)² * eps / (8 * f1 x)) <= 
eps / 4
  apply Rinv_neq_0_compat; apply prod_neq_R0; discrR || assumption.x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (f1 x / (f2 x * f2 (x + h))) <
Rabs (2 * (f1 x / (f2 x * f2 x)))
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * (f1 x / (f2 x * f2 x))) *
Rabs ((f2 x)² * eps / (8 * f1 x)) <= 
eps / 4
  unfold Rdiv.x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (f1 x * / (f2 x * f2 (x + h))) <
Rabs (2 * (f1 x * / (f2 x * f2 x)))
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * (f1 x / (f2 x * f2 x))) *
Rabs ((f2 x)² * eps / (8 * f1 x)) <= 
eps / 4
  repeat rewrite Rinv_mult_distr; try assumption.x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (f1 x * (/ f2 x * / f2 (x + h))) <
Rabs (2 * (f1 x * (/ f2 x * / f2 x)))
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * (f1 x / (f2 x * f2 x))) *
Rabs ((f2 x)² * eps / (8 * f1 x)) <= 
eps / 4
  repeat rewrite Rabs_mult.x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (f1 x) * (Rabs (/ f2 x) * Rabs (/ f2 (x + h))) <
Rabs 2 *
(Rabs (f1 x) * (Rabs (/ f2 x) * Rabs (/ f2 x)))
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * (f1 x / (f2 x * f2 x))) *
Rabs ((f2 x)² * eps / (8 * f1 x)) <= 
eps / 4
  replace (Rabs 2) with 2.x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (f1 x) * (Rabs (/ f2 x) * Rabs (/ f2 (x + h))) <
2 * (Rabs (f1 x) * (Rabs (/ f2 x) * Rabs (/ f2 x)))
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 = Rabs 2
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * (f1 x / (f2 x * f2 x))) *
Rabs ((f2 x)² * eps / (8 * f1 x)) <= 
eps / 4
  rewrite (Rmult_comm 2).x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (f1 x) * (Rabs (/ f2 x) * Rabs (/ f2 (x + h))) <
Rabs (f1 x) * (Rabs (/ f2 x) * Rabs (/ f2 x)) * 2
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 = Rabs 2
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * (f1 x / (f2 x * f2 x))) *
Rabs ((f2 x)² * eps / (8 * f1 x)) <= 
eps / 4
  replace (Rabs (f1 x) * (Rabs (/ f2 x) * Rabs (/ f2 x)) * 2) with
  (Rabs (f1 x) * (Rabs (/ f2 x) * (Rabs (/ f2 x) * 2)));
  [ idtac | ring ].x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (f1 x) * (Rabs (/ f2 x) * Rabs (/ f2 (x + h))) <
Rabs (f1 x) * (Rabs (/ f2 x) * (Rabs (/ f2 x) * 2))
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 = Rabs 2
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * (f1 x / (f2 x * f2 x))) *
Rabs ((f2 x)² * eps / (8 * f1 x)) <= 
eps / 4
  repeat apply Rmult_lt_compat_l.x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
0 < Rabs (f1 x)
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
0 < Rabs (/ f2 x)
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (/ f2 (x + h)) < Rabs (/ f2 x) * 2
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 = Rabs 2
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * (f1 x / (f2 x * f2 x))) *
Rabs ((f2 x)² * eps / (8 * f1 x)) <= 
eps / 4
  apply Rabs_pos_lt; assumption.x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
0 < Rabs (/ f2 x)
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (/ f2 (x + h)) < Rabs (/ f2 x) * 2
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 = Rabs 2
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * (f1 x / (f2 x * f2 x))) *
Rabs ((f2 x)² * eps / (8 * f1 x)) <= 
eps / 4
  apply Rabs_pos_lt; apply Rinv_neq_0_compat; assumption.x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (/ f2 (x + h)) < Rabs (/ f2 x) * 2
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 = Rabs 2
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * (f1 x / (f2 x * f2 x))) *
Rabs ((f2 x)² * eps / (8 * f1 x)) <= 
eps / 4
  repeat rewrite Rabs_Rinv; assumption || idtac.x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
/ Rabs (f2 (x + h)) < / Rabs (f2 x) * 2
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 = Rabs 2
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * (f1 x / (f2 x * f2 x))) *
Rabs ((f2 x)² * eps / (8 * f1 x)) <= 
eps / 4
  rewrite <- (Rmult_comm 2).x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
/ Rabs (f2 (x + h)) < 2 * / Rabs (f2 x)
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 = Rabs 2
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * (f1 x / (f2 x * f2 x))) *
Rabs ((f2 x)² * eps / (8 * f1 x)) <= 
eps / 4
  unfold Rdiv in H9; exact H9.x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 = Rabs 2
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * (f1 x / (f2 x * f2 x))) *
Rabs ((f2 x)² * eps / (8 * f1 x)) <= 
eps / 4
  symmetry ; apply Rabs_right; left; prove_sup0.x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * (f1 x / (f2 x * f2 x))) *
Rabs ((f2 x)² * eps / (8 * f1 x)) <= eps / 4
  right.x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * (f1 x / (f2 x * f2 x))) *
Rabs ((f2 x)² * eps / (8 * f1 x)) = eps / 4
  unfold Rsqr, Rdiv.x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * (f1 x * / (f2 x * f2 x))) *
Rabs (f2 x * f2 x * eps * / (8 * f1 x)) = eps * / 4
  rewrite Rinv_mult_distr; try assumption || discrR.x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * (f1 x * (/ f2 x * / f2 x))) *
Rabs (f2 x * f2 x * eps * / (8 * f1 x)) = eps * / 4
  rewrite Rinv_mult_distr; try assumption || discrR.x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * (f1 x * (/ f2 x * / f2 x))) *
Rabs (f2 x * f2 x * eps * (/ 8 * / f1 x)) = eps * / 4
  repeat rewrite Rabs_mult.x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs 2 *
(Rabs (f1 x) * (Rabs (/ f2 x) * Rabs (/ f2 x))) *
(Rabs (f2 x) * Rabs (f2 x) * Rabs eps *
 (Rabs (/ 8) * Rabs (/ f1 x))) = eps * / 4
  repeat rewrite Rabs_Rinv; try assumption || discrR.x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs 2 *
(Rabs (f1 x) * (/ Rabs (f2 x) * / Rabs (f2 x))) *
(Rabs (f2 x) * Rabs (f2 x) * Rabs eps *
 (/ Rabs 8 * / Rabs (f1 x))) = eps * / 4
  replace (Rabs eps) with eps.x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs 2 *
(Rabs (f1 x) * (/ Rabs (f2 x) * / Rabs (f2 x))) *
(Rabs (f2 x) * Rabs (f2 x) * eps *
 (/ Rabs 8 * / Rabs (f1 x))) = eps * / 4
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
eps = Rabs eps
  replace (Rabs 8) with 8.x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs 2 *
(Rabs (f1 x) * (/ Rabs (f2 x) * / Rabs (f2 x))) *
(Rabs (f2 x) * Rabs (f2 x) * eps *
 (/ 8 * / Rabs (f1 x))) = eps * / 4
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
8 = Rabs 8
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
eps = Rabs eps
  replace (Rabs 2) with 2.x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 * (Rabs (f1 x) * (/ Rabs (f2 x) * / Rabs (f2 x))) *
(Rabs (f2 x) * Rabs (f2 x) * eps *
 (/ 8 * / Rabs (f1 x))) = eps * / 4
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 = Rabs 2
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
8 = Rabs 8
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
eps = Rabs eps
  replace 8 with (4 * 2); [ idtac | ring ].x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 * (Rabs (f1 x) * (/ Rabs (f2 x) * / Rabs (f2 x))) *
(Rabs (f2 x) * Rabs (f2 x) * eps *
 (/ (4 * 2) * / Rabs (f1 x))) = eps * / 4
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 = Rabs 2
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
8 = Rabs 8
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
eps = Rabs eps
  rewrite Rinv_mult_distr; discrR.x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 * (Rabs (f1 x) * (/ Rabs (f2 x) * / Rabs (f2 x))) *
(Rabs (f2 x) * Rabs (f2 x) * eps *
 (/ 4 * / 2 * / Rabs (f1 x))) = eps * / 4
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 = Rabs 2
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
8 = Rabs 8
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
eps = Rabs eps
  replace
  (2 * (Rabs (f1 x) * (/ Rabs (f2 x) * / Rabs (f2 x))) *
    (Rabs (f2 x) * Rabs (f2 x) * eps * (/ 4 * / 2 * / Rabs (f1 x)))) with
  (eps * / 4 * (Rabs (f2 x) * / Rabs (f2 x)) * (Rabs (f2 x) * / Rabs (f2 x)) *
    (Rabs (f1 x) * / Rabs (f1 x)) * (2 * / 2)); [ idtac | ring ].x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
eps * / 4 * (Rabs (f2 x) * / Rabs (f2 x)) *
(Rabs (f2 x) * / Rabs (f2 x)) *
(Rabs (f1 x) * / Rabs (f1 x)) * (2 * / 2) = eps * / 4
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 = Rabs 2
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
8 = Rabs 8
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
eps = Rabs eps
  repeat rewrite <- Rinv_r_sym; try discrR || (apply Rabs_no_R0; assumption).x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
eps * / 4 * 1 * 1 * 1 * 1 = eps * / 4
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 = Rabs 2
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
8 = Rabs 8
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
eps = Rabs eps
  ring.x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 = Rabs 2
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
8 = Rabs 8
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
eps = Rabs eps
  symmetry ; apply Rabs_right; left; prove_sup0.x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
8 = Rabs 8
x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
eps = Rabs eps
  symmetry ; apply Rabs_right; left; prove_sup.x, h, eps, l2, alp_f2:R
eps_f2, alp_f2d:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < alp_f2d ->
Rabs ((f2 (x + h0) - f2 x) / h0 - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2d
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs ((f2 x)² * eps / (8 * f1 x))
H9:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
eps = Rabs eps
  symmetry ; apply Rabs_right; left; assumption.
Qed.

Lemma maj_term4 :
  forall (x h eps l2 alp_f2 alp_f2c:R) (eps_f2:posreal)
    (f1 f2:R -> R),
    0 < eps ->
    f2 x <> 0 ->
    f2 (x + h) <> 0 ->
    (forall a:R,
      Rabs a < alp_f2c ->
      Rabs (f2 (x + a) - f2 x) <
      Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))) ->
    (forall a:R,
      Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) ->
    h <> 0 ->
    Rabs h < alp_f2c ->
    Rabs h < Rmin eps_f2 alp_f2 ->
    f1 x <> 0 ->
    l2 <> 0 ->
    Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x)) <
    eps / 4.forall (x h eps l2 alp_f2 alp_f2c : R)
  (eps_f2 : posreal) (f1 f2 : R -> R),
0 < eps ->
f2 x <> 0 ->
f2 (x + h) <> 0 ->
(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 < Rmin eps_f2 alp_f2 ->
 / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) ->
h <> 0 ->
Rabs h < alp_f2c ->
Rabs h < Rmin eps_f2 alp_f2 ->
f1 x <> 0 ->
l2 <> 0 ->
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps / 4
Proof.forall (x h eps l2 alp_f2 alp_f2c : R)
  (eps_f2 : posreal) (f1 f2 : R -> R),
0 < eps ->
f2 x <> 0 ->
f2 (x + h) <> 0 ->
(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 < Rmin eps_f2 alp_f2 ->
 / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) ->
h <> 0 ->
Rabs h < alp_f2c ->
Rabs h < Rmin eps_f2 alp_f2 ->
f1 x <> 0 ->
l2 <> 0 ->
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps / 4
  intros.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps / 4
  assert (H9 := H2 h H5).x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - 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
  assert (H10 := H3 h H6).x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) < eps / 4
  apply Rle_lt_trans with
    (Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h))) *
      Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))).x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs
  (l2 * f1 x / ((f2 x)² * f2 (x + h)) *
   (f2 (x + h) - f2 x)) <=
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) <
eps / 4
  rewrite Rabs_mult.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs (f2 (x + h) - f2 x) <=
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) <
eps / 4
  apply Rmult_le_compat_l.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
0 <= Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h)))
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (f2 (x + h) - f2 x) <=
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) <
eps / 4
  apply Rabs_pos.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (f2 (x + h) - f2 x) <=
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) <
eps / 4
  left; apply H9.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) <
eps / 4
  apply Rlt_le_trans with
    (Rabs (2 * l2 * (f1 x / (Rsqr (f2 x) * f2 x))) *
      Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))).x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) *
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) <
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x))) *
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x))) *
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) <=
eps / 4
  apply Rmult_lt_compat_r.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
0 < Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) <
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x)))
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x))) *
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) <=
eps / 4
  apply Rabs_pos_lt.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
(f2 x)² * f2 x * eps / (8 * f1 x * l2) <> 0
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) <
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x)))
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x))) *
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) <=
eps / 4
  unfold Rdiv; unfold Rsqr; repeat apply prod_neq_R0;
    assumption || idtac.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
eps <> 0
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
/ (8 * f1 x * l2) <> 0
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) <
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x)))
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x))) *
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) <=
eps / 4
  red; intro H11; rewrite H11 in H; elim (Rlt_irrefl _ H).x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
/ (8 * f1 x * l2) <> 0
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) <
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x)))
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x))) *
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) <=
eps / 4
  apply Rinv_neq_0_compat; apply prod_neq_R0.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
8 * f1 x <> 0
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
l2 <> 0
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) <
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x)))
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x))) *
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) <=
eps / 4
  apply prod_neq_R0.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
8 <> 0
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
f1 x <> 0
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
l2 <> 0
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) <
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x)))
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x))) *
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) <=
eps / 4
  discrR.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
f1 x <> 0
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
l2 <> 0
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) <
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x)))
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x))) *
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) <=
eps / 4
  assumption.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
l2 <> 0
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) <
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x)))
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x))) *
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) <=
eps / 4
  assumption.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (l2 * f1 x / ((f2 x)² * f2 (x + h))) <
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x)))
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x))) *
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) <=
eps / 4
  unfold Rdiv.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (l2 * f1 x * / ((f2 x)² * f2 (x + h))) <
Rabs (2 * l2 * (f1 x * / ((f2 x)² * f2 x)))
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x))) *
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) <=
eps / 4
  repeat rewrite Rinv_mult_distr;
    try assumption || (unfold Rsqr; apply prod_neq_R0; assumption).x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (l2 * f1 x * (/ (f2 x)² * / f2 (x + h))) <
Rabs (2 * l2 * (f1 x * (/ (f2 x)² * / f2 x)))
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x))) *
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) <=
eps / 4
  repeat rewrite Rabs_mult.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs l2 * Rabs (f1 x) *
(Rabs (/ (f2 x)²) * Rabs (/ f2 (x + h))) <
Rabs 2 * Rabs l2 *
(Rabs (f1 x) * (Rabs (/ (f2 x)²) * Rabs (/ f2 x)))
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x))) *
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) <=
eps / 4
  replace (Rabs 2) with 2.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs l2 * Rabs (f1 x) *
(Rabs (/ (f2 x)²) * Rabs (/ f2 (x + h))) <
2 * Rabs l2 *
(Rabs (f1 x) * (Rabs (/ (f2 x)²) * Rabs (/ f2 x)))
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 = Rabs 2
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x))) *
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) <=
eps / 4
  replace
  (2 * Rabs l2 * (Rabs (f1 x) * (Rabs (/ Rsqr (f2 x)) * Rabs (/ f2 x)))) with
  (Rabs l2 * (Rabs (f1 x) * (Rabs (/ Rsqr (f2 x)) * (Rabs (/ f2 x) * 2))));
  [ idtac | ring ].x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs l2 * Rabs (f1 x) *
(Rabs (/ (f2 x)²) * Rabs (/ f2 (x + h))) <
Rabs l2 *
(Rabs (f1 x) *
 (Rabs (/ (f2 x)²) * (Rabs (/ f2 x) * 2)))
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 = Rabs 2
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x))) *
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) <=
eps / 4
  replace
  (Rabs l2 * Rabs (f1 x) * (Rabs (/ Rsqr (f2 x)) * Rabs (/ f2 (x + h)))) with
  (Rabs l2 * (Rabs (f1 x) * (Rabs (/ Rsqr (f2 x)) * Rabs (/ f2 (x + h)))));
  [ idtac | ring ].x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs l2 *
(Rabs (f1 x) *
 (Rabs (/ (f2 x)²) * Rabs (/ f2 (x + h)))) <
Rabs l2 *
(Rabs (f1 x) *
 (Rabs (/ (f2 x)²) * (Rabs (/ f2 x) * 2)))
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 = Rabs 2
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x))) *
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) <=
eps / 4
  repeat apply Rmult_lt_compat_l.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
0 < Rabs l2
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
0 < Rabs (f1 x)
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
0 < Rabs (/ (f2 x)²)
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (/ f2 (x + h)) < Rabs (/ f2 x) * 2
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 = Rabs 2
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x))) *
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) <=
eps / 4
  apply Rabs_pos_lt; assumption.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
0 < Rabs (f1 x)
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
0 < Rabs (/ (f2 x)²)
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (/ f2 (x + h)) < Rabs (/ f2 x) * 2
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 = Rabs 2
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x))) *
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) <=
eps / 4
  apply Rabs_pos_lt; assumption.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
0 < Rabs (/ (f2 x)²)
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (/ f2 (x + h)) < Rabs (/ f2 x) * 2
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 = Rabs 2
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x))) *
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) <=
eps / 4
  apply Rabs_pos_lt; apply Rinv_neq_0_compat; unfold Rsqr;
    apply prod_neq_R0; assumption.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (/ f2 (x + h)) < Rabs (/ f2 x) * 2
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 = Rabs 2
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x))) *
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) <=
eps / 4
  repeat rewrite Rabs_Rinv; [ idtac | assumption | assumption ].x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
/ Rabs (f2 (x + h)) < / Rabs (f2 x) * 2
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 = Rabs 2
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x))) *
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) <=
eps / 4
  rewrite <- (Rmult_comm 2).x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
/ Rabs (f2 (x + h)) < 2 * / Rabs (f2 x)
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 = Rabs 2
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x))) *
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) <=
eps / 4
  unfold Rdiv in H10; exact H10.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 = Rabs 2
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x))) *
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) <=
eps / 4
  symmetry ; apply Rabs_right; left; prove_sup0.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * l2 * (f1 x / ((f2 x)² * f2 x))) *
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2)) <=
eps / 4
  right; unfold Rsqr, Rdiv.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * l2 * (f1 x * / (f2 x * f2 x * f2 x))) *
Rabs (f2 x * f2 x * f2 x * eps * / (8 * f1 x * l2)) =
eps * / 4
  rewrite Rinv_mult_distr; try assumption || discrR.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * l2 * (f1 x * (/ (f2 x * f2 x) * / f2 x))) *
Rabs (f2 x * f2 x * f2 x * eps * / (8 * f1 x * l2)) =
eps * / 4
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
f2 x * f2 x <> 0
  rewrite Rinv_mult_distr; try assumption || discrR.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * l2 * (f1 x * (/ f2 x * / f2 x * / f2 x))) *
Rabs (f2 x * f2 x * f2 x * eps * / (8 * f1 x * l2)) =
eps * / 4
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
f2 x * f2 x <> 0
  rewrite Rinv_mult_distr; try assumption || discrR.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * l2 * (f1 x * (/ f2 x * / f2 x * / f2 x))) *
Rabs
  (f2 x * f2 x * f2 x * eps * (/ (8 * f1 x) * / l2)) =
eps * / 4
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
8 * f1 x <> 0
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
f2 x * f2 x <> 0
  rewrite Rinv_mult_distr; try assumption || discrR.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs (2 * l2 * (f1 x * (/ f2 x * / f2 x * / f2 x))) *
Rabs
  (f2 x * f2 x * f2 x * eps * (/ 8 * / f1 x * / l2)) =
eps * / 4
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
8 * f1 x <> 0
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
f2 x * f2 x <> 0
  repeat rewrite Rabs_mult.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs 2 * Rabs l2 *
(Rabs (f1 x) *
 (Rabs (/ f2 x) * Rabs (/ f2 x) * Rabs (/ f2 x))) *
(Rabs (f2 x) * Rabs (f2 x) * Rabs (f2 x) * Rabs eps *
 (Rabs (/ 8) * Rabs (/ f1 x) * Rabs (/ l2))) =
eps * / 4
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
8 * f1 x <> 0
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
f2 x * f2 x <> 0
  repeat rewrite Rabs_Rinv; try assumption || discrR.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs 2 * Rabs l2 *
(Rabs (f1 x) *
 (/ Rabs (f2 x) * / Rabs (f2 x) * / Rabs (f2 x))) *
(Rabs (f2 x) * Rabs (f2 x) * Rabs (f2 x) * Rabs eps *
 (/ Rabs 8 * / Rabs (f1 x) * / Rabs l2)) = eps * / 4
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
8 * f1 x <> 0
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
f2 x * f2 x <> 0
  replace (Rabs eps) with eps.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs 2 * Rabs l2 *
(Rabs (f1 x) *
 (/ Rabs (f2 x) * / Rabs (f2 x) * / Rabs (f2 x))) *
(Rabs (f2 x) * Rabs (f2 x) * Rabs (f2 x) * eps *
 (/ Rabs 8 * / Rabs (f1 x) * / Rabs l2)) = eps * / 4
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
eps = Rabs eps
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
8 * f1 x <> 0
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
f2 x * f2 x <> 0
  replace (Rabs 8) with 8.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
Rabs 2 * Rabs l2 *
(Rabs (f1 x) *
 (/ Rabs (f2 x) * / Rabs (f2 x) * / Rabs (f2 x))) *
(Rabs (f2 x) * Rabs (f2 x) * Rabs (f2 x) * eps *
 (/ 8 * / Rabs (f1 x) * / Rabs l2)) = eps * / 4
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
8 = Rabs 8
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
eps = Rabs eps
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
8 * f1 x <> 0
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
f2 x * f2 x <> 0
  replace (Rabs 2) with 2.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 * Rabs l2 *
(Rabs (f1 x) *
 (/ Rabs (f2 x) * / Rabs (f2 x) * / Rabs (f2 x))) *
(Rabs (f2 x) * Rabs (f2 x) * Rabs (f2 x) * eps *
 (/ 8 * / Rabs (f1 x) * / Rabs l2)) = eps * / 4
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 = Rabs 2
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
8 = Rabs 8
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
eps = Rabs eps
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
8 * f1 x <> 0
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
f2 x * f2 x <> 0
  replace 8 with (4 * 2); [ idtac | ring ].x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 * Rabs l2 *
(Rabs (f1 x) *
 (/ Rabs (f2 x) * / Rabs (f2 x) * / Rabs (f2 x))) *
(Rabs (f2 x) * Rabs (f2 x) * Rabs (f2 x) * eps *
 (/ (4 * 2) * / Rabs (f1 x) * / Rabs l2)) = eps * / 4
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 = Rabs 2
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
8 = Rabs 8
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
eps = Rabs eps
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
8 * f1 x <> 0
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
f2 x * f2 x <> 0
  rewrite Rinv_mult_distr; discrR.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 * Rabs l2 *
(Rabs (f1 x) *
 (/ Rabs (f2 x) * / Rabs (f2 x) * / Rabs (f2 x))) *
(Rabs (f2 x) * Rabs (f2 x) * Rabs (f2 x) * eps *
 (/ 4 * / 2 * / Rabs (f1 x) * / Rabs l2)) = eps * / 4
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 = Rabs 2
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
8 = Rabs 8
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
eps = Rabs eps
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
8 * f1 x <> 0
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
f2 x * f2 x <> 0
  replace
  (2 * Rabs l2 *
    (Rabs (f1 x) * (/ Rabs (f2 x) * / Rabs (f2 x) * / Rabs (f2 x))) *
    (Rabs (f2 x) * Rabs (f2 x) * Rabs (f2 x) * eps *
      (/ 4 * / 2 * / Rabs (f1 x) * / Rabs l2))) with
  (eps * / 4 * (Rabs l2 * / Rabs l2) * (Rabs (f1 x) * / Rabs (f1 x)) *
    (Rabs (f2 x) * / Rabs (f2 x)) * (Rabs (f2 x) * / Rabs (f2 x)) *
    (Rabs (f2 x) * / Rabs (f2 x)) * (2 * / 2)); [ idtac | ring ].x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
eps * / 4 * (Rabs l2 * / Rabs l2) *
(Rabs (f1 x) * / Rabs (f1 x)) *
(Rabs (f2 x) * / Rabs (f2 x)) *
(Rabs (f2 x) * / Rabs (f2 x)) *
(Rabs (f2 x) * / Rabs (f2 x)) * (2 * / 2) = eps * / 4
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 = Rabs 2
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
8 = Rabs 8
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
eps = Rabs eps
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
8 * f1 x <> 0
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
f2 x * f2 x <> 0
  repeat rewrite <- Rinv_r_sym; try discrR || (apply Rabs_no_R0; assumption).x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
eps * / 4 * 1 * 1 * 1 * 1 * 1 * 1 = eps * / 4
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 = Rabs 2
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
8 = Rabs 8
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
eps = Rabs eps
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
8 * f1 x <> 0
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
f2 x * f2 x <> 0
  ring.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
2 = Rabs 2
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
8 = Rabs 8
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
eps = Rabs eps
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
8 * f1 x <> 0
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
f2 x * f2 x <> 0
  symmetry ; apply Rabs_right; left; prove_sup0.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
8 = Rabs 8
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
eps = Rabs eps
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
8 * f1 x <> 0
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
f2 x * f2 x <> 0
  symmetry ; apply Rabs_right; left; prove_sup.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
eps = Rabs eps
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
8 * f1 x <> 0
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
f2 x * f2 x <> 0
  symmetry ; apply Rabs_right; left; assumption.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
8 * f1 x <> 0
x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
f2 x * f2 x <> 0
  apply prod_neq_R0; assumption || discrR.x, h, eps, l2, alp_f2, alp_f2c:R
eps_f2:posreal
f1, f2:R -> R
H:0 < eps
H0:f2 x <> 0
H1:f2 (x + h) <> 0
H2:forall a : R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H3:forall a : R,
Rabs a < Rmin eps_f2 alp_f2 ->
/ Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4:h <> 0
H5:Rabs h < alp_f2c
H6:Rabs h < Rmin eps_f2 alp_f2
H7:f1 x <> 0
H8:l2 <> 0
H9:Rabs (f2 (x + h) - f2 x) <
Rabs ((f2 x)² * f2 x * eps / (8 * f1 x * l2))
H10:/ Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
f2 x * f2 x <> 0
  apply prod_neq_R0; assumption.
Qed.

Lemma D_x_no_cond : forall x a:R, a <> 0 -> D_x no_cond x (x + a).forall x a : R, a <> 0 -> D_x no_cond x (x + a)
Proof.forall x a : R, a <> 0 -> D_x no_cond x (x + a)
  intros.x, a:R
H:a <> 0
D_x no_cond x (x + a)
  unfold D_x, no_cond.x, a:R
H:a <> 0
True /\ x <> x + a
  split.x, a:R
H:a <> 0
True
x, a:R
H:a <> 0
x <> x + a
  trivial.x, a:R
H:a <> 0
x <> x + a
  apply Rminus_not_eq.x, a:R
H:a <> 0
x - (x + a) <> 0
  unfold Rminus.x, a:R
H:a <> 0
x + - (x + a) <> 0
  rewrite Ropp_plus_distr.x, a:R
H:a <> 0
x + (- x + - a) <> 0
  rewrite <- Rplus_assoc.x, a:R
H:a <> 0
x + - x + - a <> 0
  rewrite Rplus_opp_r.x, a:R
H:a <> 0
0 + - a <> 0
  rewrite Rplus_0_l.x, a:R
H:a <> 0
- a <> 0
  apply Ropp_neq_0_compat; assumption.
Qed.

Lemma Rabs_4 :
  forall a b c d:R, Rabs (a + b + c + d) <= Rabs a + Rabs b + Rabs c + Rabs d.forall a b c d : R,
Rabs (a + b + c + d) <=
Rabs a + Rabs b + Rabs c + Rabs d
Proof.forall a b c d : R,
Rabs (a + b + c + d) <=
Rabs a + Rabs b + Rabs c + Rabs d
  intros.a, b, c, d:R
Rabs (a + b + c + d) <=
Rabs a + Rabs b + Rabs c + Rabs d
  apply Rle_trans with (Rabs (a + b) + Rabs (c + d)).a, b, c, d:R
Rabs (a + b + c + d) <= Rabs (a + b) + Rabs (c + d)
a, b, c, d:R
Rabs (a + b) + Rabs (c + d) <=
Rabs a + Rabs b + Rabs c + Rabs d
  replace (a + b + c + d) with (a + b + (c + d)); [ apply Rabs_triang | ring ].a, b, c, d:R
Rabs (a + b) + Rabs (c + d) <=
Rabs a + Rabs b + Rabs c + Rabs d
  apply Rle_trans with (Rabs a + Rabs b + Rabs (c + d)).a, b, c, d:R
Rabs (a + b) + Rabs (c + d) <=
Rabs a + Rabs b + Rabs (c + d)
a, b, c, d:R
Rabs a + Rabs b + Rabs (c + d) <=
Rabs a + Rabs b + Rabs c + Rabs d
  apply Rplus_le_compat_r.a, b, c, d:R
Rabs (a + b) <= Rabs a + Rabs b
a, b, c, d:R
Rabs a + Rabs b + Rabs (c + d) <=
Rabs a + Rabs b + Rabs c + Rabs d
  apply Rabs_triang.a, b, c, d:R
Rabs a + Rabs b + Rabs (c + d) <=
Rabs a + Rabs b + Rabs c + Rabs d
  repeat rewrite Rplus_assoc; repeat apply Rplus_le_compat_l.a, b, c, d:R
Rabs (c + d) <= Rabs c + Rabs d
  apply Rabs_triang.
Qed.

Lemma Rlt_4 :
  forall a b c d e f g h:R,
    a < b -> c < d -> e < f -> g < h -> a + c + e + g < b + d + f + h.forall a b c d e f g h : R,
a < b ->
c < d ->
e < f -> g < h -> a + c + e + g < b + d + f + h
Proof.forall a b c d e f g h : R,
a < b ->
c < d ->
e < f -> g < h -> a + c + e + g < b + d + f + h
  intros; apply Rlt_trans with (b + c + e + g).a, b, c, d, e, f, g, h:R
H:a < b
H0:c < d
H1:e < f
H2:g < h
a + c + e + g < b + c + e + g
a, b, c, d, e, f, g, h:R
H:a < b
H0:c < d
H1:e < f
H2:g < h
b + c + e + g < b + d + f + h
  repeat apply Rplus_lt_compat_r; assumption.a, b, c, d, e, f, g, h:R
H:a < b
H0:c < d
H1:e < f
H2:g < h
b + c + e + g < b + d + f + h
  repeat rewrite Rplus_assoc; apply Rplus_lt_compat_l.a, b, c, d, e, f, g, h:R
H:a < b
H0:c < d
H1:e < f
H2:g < h
c + (e + g) < d + (f + h)
  apply Rlt_trans with (d + e + g).a, b, c, d, e, f, g, h:R
H:a < b
H0:c < d
H1:e < f
H2:g < h
c + (e + g) < d + e + g
a, b, c, d, e, f, g, h:R
H:a < b
H0:c < d
H1:e < f
H2:g < h
d + e + g < d + (f + h)
  rewrite Rplus_assoc; apply Rplus_lt_compat_r; assumption.a, b, c, d, e, f, g, h:R
H:a < b
H0:c < d
H1:e < f
H2:g < h
d + e + g < d + (f + h)
  rewrite Rplus_assoc; apply Rplus_lt_compat_l; apply Rlt_trans with (f + g).a, b, c, d, e, f, g, h:R
H:a < b
H0:c < d
H1:e < f
H2:g < h
e + g < f + g
a, b, c, d, e, f, g, h:R
H:a < b
H0:c < d
H1:e < f
H2:g < h
f + g < f + h
  apply Rplus_lt_compat_r; assumption.a, b, c, d, e, f, g, h:R
H:a < b
H0:c < d
H1:e < f
H2:g < h
f + g < f + h
  apply Rplus_lt_compat_l; assumption.
Qed.

(* begin hide *)
Notation Rmin_2 := Rmin_glb_lt (only parsing).
(* end hide *)

Lemma quadruple : forall x:R, 4 * x = x + x + x + x.forall x : R, 4 * x = x + x + x + x
Proof.forall x : R, 4 * x = x + x + x + x
  intro; ring.
Qed.

Lemma quadruple_var : forall x:R, x = x / 4 + x / 4 + x / 4 + x / 4.forall x : R, x = x / 4 + x / 4 + x / 4 + x / 4
Proof.forall x : R, x = x / 4 + x / 4 + x / 4 + x / 4
  intro; rewrite <- quadruple.x:R
x = 4 * (x / 4)
  unfold Rdiv; rewrite <- Rmult_assoc; rewrite Rinv_r_simpl_m; discrR.x:R
x = x
  reflexivity.
Qed.

(**********)
Lemma continuous_neq_0 :
  forall (f:R -> R) (x0:R),
    continuity_pt f x0 ->
    f x0 <> 0 ->
    exists eps : posreal, (forall h:R, Rabs h < eps -> f (x0 + h) <> 0).forall (f : R -> R) (x0 : R),
continuity_pt f x0 ->
f x0 <> 0 ->
exists eps : posreal,
  forall h : R, Rabs h < eps -> f (x0 + h) <> 0
Proof.forall (f : R -> R) (x0 : R),
continuity_pt f x0 ->
f x0 <> 0 ->
exists eps : posreal,
  forall h : R, Rabs h < eps -> f (x0 + h) <> 0
  intros; unfold continuity_pt in H; unfold continue_in in H;
    unfold limit1_in in H; unfold limit_in in H; elim (H (Rabs (f x0 / 2))).f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
forall x : R,
x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2)) ->
exists eps : posreal,
  forall h : R, Rabs h < eps -> f (x0 + h) <> 0
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
Rabs (f x0 / 2) > 0
  intros; elim H1; intros.f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
exists eps : posreal,
  forall h : R, Rabs h < eps -> f (x0 + h) <> 0
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
Rabs (f x0 / 2) > 0
  exists (mkposreal x H2).f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
forall h : R,
Rabs h < {| pos := x; cond_pos := H2 |} ->
f (x0 + h) <> 0
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
Rabs (f x0 / 2) > 0
  intros; assert (H5 := H3 (x0 + h)).f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
f (x0 + h) <> 0
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
Rabs (f x0 / 2) > 0
  cut
    (dist R_met (x0 + h) x0 < x ->
      dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)).f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
(dist R_met (x0 + h) x0 < x ->
 dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)) ->
f (x0 + h) <> 0
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
Rabs (f x0 / 2) > 0
  unfold dist; simpl; unfold R_dist;
    replace (x0 + h - x0) with h.f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
(Rabs h < x ->
 Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)) ->
f (x0 + h) <> 0
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
h = x0 + h - x0
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
Rabs (f x0 / 2) > 0
  intros; assert (H7 := H6 H4).f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
f (x0 + h) <> 0
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
h = x0 + h - x0
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
Rabs (f x0 / 2) > 0
  red; intro.f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H8:f (x0 + h) = 0
False
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
h = x0 + h - x0
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
Rabs (f x0 / 2) > 0
  rewrite H8 in H7; unfold Rminus in H7; rewrite Rplus_0_l in H7;
    rewrite Rabs_Ropp in H7; unfold Rdiv in H7; rewrite Rabs_mult in H7;
      pattern (Rabs (f x0)) at 1 in H7; rewrite <- Rmult_1_r in H7.f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
False
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
h = x0 + h - x0
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
Rabs (f x0 / 2) > 0
  cut (0 < Rabs (f x0)).f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
0 < Rabs (f x0) -> False
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
0 < Rabs (f x0)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
h = x0 + h - x0
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
Rabs (f x0 / 2) > 0
  intro; assert (H10 := Rmult_lt_reg_l _ _ _ H9 H7).f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
H9:0 < Rabs (f x0)
H10:1 < Rabs (/ 2)
False
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
0 < Rabs (f x0)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
h = x0 + h - x0
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
Rabs (f x0 / 2) > 0
  cut (Rabs (/ 2) = / 2).f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
H9:0 < Rabs (f x0)
H10:1 < Rabs (/ 2)
Rabs (/ 2) = / 2 -> False
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
H9:0 < Rabs (f x0)
H10:1 < Rabs (/ 2)
Rabs (/ 2) = / 2
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
0 < Rabs (f x0)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
h = x0 + h - x0
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
Rabs (f x0 / 2) > 0
  assert (Hyp : 0 < 2).f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
H9:0 < Rabs (f x0)
H10:1 < Rabs (/ 2)
0 < 2
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
H9:0 < Rabs (f x0)
H10:1 < Rabs (/ 2)
Hyp:0 < 2
Rabs (/ 2) = / 2 -> False
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
H9:0 < Rabs (f x0)
H10:1 < Rabs (/ 2)
Rabs (/ 2) = / 2
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
0 < Rabs (f x0)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
h = x0 + h - x0
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
Rabs (f x0 / 2) > 0
  prove_sup0.f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
H9:0 < Rabs (f x0)
H10:1 < Rabs (/ 2)
Hyp:0 < 2
Rabs (/ 2) = / 2 -> False
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
H9:0 < Rabs (f x0)
H10:1 < Rabs (/ 2)
Rabs (/ 2) = / 2
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
0 < Rabs (f x0)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
h = x0 + h - x0
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
Rabs (f x0 / 2) > 0
  intro; rewrite H11 in H10; assert (H12 := Rmult_lt_compat_l 2 _ _ Hyp H10);
    rewrite Rmult_1_r in H12; rewrite <- Rinv_r_sym in H12;
      [ idtac | discrR ].f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
H9:0 < Rabs (f x0)
H10:1 < / 2
Hyp:0 < 2
H11:Rabs (/ 2) = / 2
H12:2 < 1
False
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
H9:0 < Rabs (f x0)
H10:1 < Rabs (/ 2)
Rabs (/ 2) = / 2
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
0 < Rabs (f x0)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
h = x0 + h - x0
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
Rabs (f x0 / 2) > 0
  now apply lt_IZR in H12.f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
H9:0 < Rabs (f x0)
H10:1 < Rabs (/ 2)
Rabs (/ 2) = / 2
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
0 < Rabs (f x0)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
h = x0 + h - x0
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
Rabs (f x0 / 2) > 0
  unfold Rabs; case (Rcase_abs (/ 2)) as [Hlt|Hge].f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
H9:0 < Rabs (f x0)
H10:1 < Rabs (/ 2)
Hlt:/ 2 < 0
- / 2 = / 2
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
H9:0 < Rabs (f x0)
H10:1 < Rabs (/ 2)
Hge:/ 2 >= 0
/ 2 = / 2
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
0 < Rabs (f x0)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
h = x0 + h - x0
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
Rabs (f x0 / 2) > 0
  assert (Hyp : 0 < 2).f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
H9:0 < Rabs (f x0)
H10:1 < Rabs (/ 2)
Hlt:/ 2 < 0
0 < 2
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
H9:0 < Rabs (f x0)
H10:1 < Rabs (/ 2)
Hlt:/ 2 < 0
Hyp:0 < 2
- / 2 = / 2
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
H9:0 < Rabs (f x0)
H10:1 < Rabs (/ 2)
Hge:/ 2 >= 0
/ 2 = / 2
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
0 < Rabs (f x0)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
h = x0 + h - x0
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
Rabs (f x0 / 2) > 0
  prove_sup0.f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
H9:0 < Rabs (f x0)
H10:1 < Rabs (/ 2)
Hlt:/ 2 < 0
Hyp:0 < 2
- / 2 = / 2
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
H9:0 < Rabs (f x0)
H10:1 < Rabs (/ 2)
Hge:/ 2 >= 0
/ 2 = / 2
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
0 < Rabs (f x0)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
h = x0 + h - x0
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
Rabs (f x0 / 2) > 0
  assert (H11 := Rmult_lt_compat_l 2 _ _ Hyp Hlt); rewrite Rmult_0_r in H11;
    rewrite <- Rinv_r_sym in H11; [ idtac | discrR ].f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
H9:0 < Rabs (f x0)
H10:1 < Rabs (/ 2)
Hlt:/ 2 < 0
Hyp:0 < 2
H11:1 < 0
- / 2 = / 2
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
H9:0 < Rabs (f x0)
H10:1 < Rabs (/ 2)
Hge:/ 2 >= 0
/ 2 = / 2
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
0 < Rabs (f x0)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
h = x0 + h - x0
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
Rabs (f x0 / 2) > 0
  elim (Rlt_irrefl 0 (Rlt_trans _ _ _ Rlt_0_1 H11)).f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
H9:0 < Rabs (f x0)
H10:1 < Rabs (/ 2)
Hge:/ 2 >= 0
/ 2 = / 2
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
0 < Rabs (f x0)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
h = x0 + h - x0
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
Rabs (f x0 / 2) > 0
  reflexivity.f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:Rabs h < x ->
Rabs (f (x0 + h) - f x0) < Rabs (f x0 / 2)
H7:Rabs (f x0) * 1 < Rabs (f x0) * Rabs (/ 2)
H8:f (x0 + h) = 0
0 < Rabs (f x0)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
h = x0 + h - x0
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
Rabs (f x0 / 2) > 0
  apply (Rabs_pos_lt _ H0).f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
h = x0 + h - x0
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
Rabs (f x0 / 2) > 0
  ring.f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
Rabs (f x0 / 2) > 0
  assert (H6 := Req_dec x0 (x0 + h)); elim H6; intro.f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:x0 = x0 + h \/ x0 <> x0 + h
H7:x0 = x0 + h
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:x0 = x0 + h \/ x0 <> x0 + h
H7:x0 <> x0 + h
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
Rabs (f x0 / 2) > 0
  intro; rewrite <- H7.f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:x0 = x0 + h \/ x0 <> x0 + h
H7:x0 = x0 + h
H8:dist R_met (x0 + h) x0 < x
dist R_met (f x0) (f x0) < Rabs (f x0 / 2)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:x0 = x0 + h \/ x0 <> x0 + h
H7:x0 <> x0 + h
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
Rabs (f x0 / 2) > 0 unfold R_met, dist; unfold R_dist;
    unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0;
      apply Rabs_pos_lt.f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:x0 = x0 + h \/ x0 <> x0 + h
H7:x0 = x0 + h
H8:dist R_met (x0 + h) x0 < x
f x0 / 2 <> 0
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:x0 = x0 + h \/ x0 <> x0 + h
H7:x0 <> x0 + h
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
Rabs (f x0 / 2) > 0
  unfold Rdiv; apply prod_neq_R0;
    [ assumption | apply Rinv_neq_0_compat; discrR ].f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:x0 = x0 + h \/ x0 <> x0 + h
H7:x0 <> x0 + h
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
Rabs (f x0 / 2) > 0
  intro; apply H5.f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:x0 = x0 + h \/ x0 <> x0 + h
H7:x0 <> x0 + h
H8:dist R_met (x0 + h) x0 < x
D_x no_cond x0 (x0 + h) /\ dist R_met (x0 + h) x0 < x
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
Rabs (f x0 / 2) > 0
  split.f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:x0 = x0 + h \/ x0 <> x0 + h
H7:x0 <> x0 + h
H8:dist R_met (x0 + h) x0 < x
D_x no_cond x0 (x0 + h)
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:x0 = x0 + h \/ x0 <> x0 + h
H7:x0 <> x0 + h
H8:dist R_met (x0 + h) x0 < x
dist R_met (x0 + h) x0 < x
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
Rabs (f x0 / 2) > 0
  unfold D_x, no_cond.f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:x0 = x0 + h \/ x0 <> x0 + h
H7:x0 <> x0 + h
H8:dist R_met (x0 + h) x0 < x
True /\ x0 <> x0 + h
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:x0 = x0 + h \/ x0 <> x0 + h
H7:x0 <> x0 + h
H8:dist R_met (x0 + h) x0 < x
dist R_met (x0 + h) x0 < x
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
Rabs (f x0 / 2) > 0
  split; trivial || assumption.f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   D_x no_cond x0 x1 /\ dist R_met x1 x0 < alp ->
   dist R_met (f x1) (f x0) < eps)
H0:f x0 <> 0
x:R
H1:x > 0 /\
(forall x1 : Base R_met,
 D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) (f x0) < Rabs (f x0 / 2))
H2:x > 0
H3:forall x1 : Base R_met,
D_x no_cond x0 x1 /\ dist R_met x1 x0 < x ->
dist R_met (f x1) (f x0) < Rabs (f x0 / 2)
h:R
H4:Rabs h < {| pos := x; cond_pos := H2 |}
H5:D_x no_cond x0 (x0 + h) /\
dist R_met (x0 + h) x0 < x ->
dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)
H6:x0 = x0 + h \/ x0 <> x0 + h
H7:x0 <> x0 + h
H8:dist R_met (x0 + h) x0 < x
dist R_met (x0 + h) x0 < x
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
Rabs (f x0 / 2) > 0
  assumption.f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
Rabs (f x0 / 2) > 0
  change (0 < Rabs (f x0 / 2)).f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
0 < Rabs (f x0 / 2)
  apply Rabs_pos_lt; unfold Rdiv; apply prod_neq_R0.f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
f x0 <> 0
f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
/ 2 <> 0
  assumption.f:R -> R
x0:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D_x no_cond x0 x /\ dist R_met x x0 < alp ->
   dist R_met (f x) (f x0) < eps)
H0:f x0 <> 0
/ 2 <> 0
  apply Rinv_neq_0_compat; discrR.
Qed.