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(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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Require Import Rbase.
Require Import Rfunctions.
Require Import Lra.
Local Open Scope R_scope.

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(*********)
Lemma eps2_Rgt_R0 : forall eps:R, eps > 0 -> eps * / 2 > 0.
forall eps : R, eps > 0 -> eps * / 2 > 0
Proof.
forall eps : R, eps > 0 -> eps * / 2 > 0
  intros; lra.
Qed.

(*********)
Lemma eps2 : forall eps:R, eps * / 2 + eps * / 2 = eps.
forall eps : R, eps * / 2 + eps * / 2 = eps
Proof.
forall eps : R, eps * / 2 + eps * / 2 = eps
  intro esp.
esp:R
esp * / 2 + esp * / 2 = esp
  apply eq_sym, double_var.
Qed.

(*********)
Lemma eps4 : forall eps:R, eps * / (2 + 2) + eps * / (2 + 2) = eps * / 2.
forall eps : R,
eps * / (2 + 2) + eps * / (2 + 2) = eps * / 2
Proof.
forall eps : R,
eps * / (2 + 2) + eps * / (2 + 2) = eps * / 2
  intro eps.
eps:R
eps * / (2 + 2) + eps * / (2 + 2) = eps * / 2
  field.
Qed.

(*********)
Lemma Rlt_eps2_eps : forall eps:R, eps > 0 -> eps * / 2 < eps.
forall eps : R, eps > 0 -> eps * / 2 < eps
Proof.
forall eps : R, eps > 0 -> eps * / 2 < eps
  intros.
eps:R
H:eps > 0
eps * / 2 < eps
  lra.
Qed.

(*********)
Lemma Rlt_eps4_eps : forall eps:R, eps > 0 -> eps * / (2 + 2) < eps.
forall eps : R, eps > 0 -> eps * / (2 + 2) < eps
Proof.
forall eps : R, eps > 0 -> eps * / (2 + 2) < eps
  intros.
eps:R
H:eps > 0
eps * / (2 + 2) < eps
  lra.
Qed.

(*********)
Lemma prop_eps : forall r:R, (forall eps:R, eps > 0 -> r < eps) -> r <= 0.
forall r : R,
(forall eps : R, eps > 0 -> r < eps) -> r <= 0
Proof.
forall r : R,
(forall eps : R, eps > 0 -> r < eps) -> r <= 0
  intros; elim (Rtotal_order r 0); intro.
r:R
H:forall eps : R, eps > 0 -> r < eps
H0:r < 0
r <= 0
r:R
H:forall eps : R, eps > 0 -> r < eps
H0:r = 0 \/ r > 0
r <= 0
  apply Rlt_le; assumption.
r:R
H:forall eps : R, eps > 0 -> r < eps
H0:r = 0 \/ r > 0
r <= 0
  elim H0; intro.
r:R
H:forall eps : R, eps > 0 -> r < eps
H0:r = 0 \/ r > 0
H1:r = 0
r <= 0
r:R
H:forall eps : R, eps > 0 -> r < eps
H0:r = 0 \/ r > 0
H1:r > 0
r <= 0
  apply Req_le; assumption.
r:R
H:forall eps : R, eps > 0 -> r < eps
H0:r = 0 \/ r > 0
H1:r > 0
r <= 0
  clear H0; generalize (H r H1); intro; generalize (Rlt_irrefl r); intro;
    exfalso; auto.
Qed.

(*********)
Definition mul_factor (l l':R) := / (1 + (Rabs l + Rabs l')).

(*********)
Lemma mul_factor_wd : forall l l':R, 1 + (Rabs l + Rabs l') <> 0.
forall l l' : R, 1 + (Rabs l + Rabs l') <> 0
Proof.
forall l l' : R, 1 + (Rabs l + Rabs l') <> 0
  intros; rewrite (Rplus_comm 1 (Rabs l + Rabs l')); apply tech_Rplus.
l, l':R
0 <= Rabs l + Rabs l'
l, l':R
0 < 1
  cut (Rabs (l + l') <= Rabs l + Rabs l').
l, l':R
Rabs (l + l') <= Rabs l + Rabs l' ->
0 <= Rabs l + Rabs l'
l, l':R
Rabs (l + l') <= Rabs l + Rabs l'
l, l':R
0 < 1
  cut (0 <= Rabs (l + l')).
l, l':R
0 <= Rabs (l + l') ->
Rabs (l + l') <= Rabs l + Rabs l' ->
0 <= Rabs l + Rabs l'
l, l':R
0 <= Rabs (l + l')
l, l':R
Rabs (l + l') <= Rabs l + Rabs l'
l, l':R
0 < 1
  exact (Rle_trans _ _ _).
l, l':R
0 <= Rabs (l + l')
l, l':R
Rabs (l + l') <= Rabs l + Rabs l'
l, l':R
0 < 1
  exact (Rabs_pos (l + l')).
l, l':R
Rabs (l + l') <= Rabs l + Rabs l'
l, l':R
0 < 1
  exact (Rabs_triang _ _).
l, l':R
0 < 1
  exact Rlt_0_1.
Qed.

(*********)
Lemma mul_factor_gt : forall eps l l':R, eps > 0 -> eps * mul_factor l l' > 0.
forall eps l l' : R,
eps > 0 -> eps * mul_factor l l' > 0
Proof.
forall eps l l' : R,
eps > 0 -> eps * mul_factor l l' > 0
  intros; unfold Rgt; rewrite <- (Rmult_0_r eps);
    apply Rmult_lt_compat_l.
eps, l, l':R
H:eps > 0
0 < eps
eps, l, l':R
H:eps > 0
0 < mul_factor l l'
  assumption.
eps, l, l':R
H:eps > 0
0 < mul_factor l l'
  unfold mul_factor; apply Rinv_0_lt_compat;
    cut (1 <= 1 + (Rabs l + Rabs l')).
eps, l, l':R
H:eps > 0
1 <= 1 + (Rabs l + Rabs l') ->
0 < 1 + (Rabs l + Rabs l')
eps, l, l':R
H:eps > 0
1 <= 1 + (Rabs l + Rabs l')
  cut (0 < 1).
eps, l, l':R
H:eps > 0
0 < 1 ->
1 <= 1 + (Rabs l + Rabs l') ->
0 < 1 + (Rabs l + Rabs l')
eps, l, l':R
H:eps > 0
0 < 1
eps, l, l':R
H:eps > 0
1 <= 1 + (Rabs l + Rabs l')
  exact (Rlt_le_trans _ _ _).
eps, l, l':R
H:eps > 0
0 < 1
eps, l, l':R
H:eps > 0
1 <= 1 + (Rabs l + Rabs l')
  exact Rlt_0_1.
eps, l, l':R
H:eps > 0
1 <= 1 + (Rabs l + Rabs l')
  replace (1 <= 1 + (Rabs l + Rabs l')) with (1 + 0 <= 1 + (Rabs l + Rabs l')).
eps, l, l':R
H:eps > 0
1 + 0 <= 1 + (Rabs l + Rabs l')
eps, l, l':R
H:eps > 0
(1 + 0 <= 1 + (Rabs l + Rabs l')) =
(1 <= 1 + (Rabs l + Rabs l'))
  apply Rplus_le_compat_l.
eps, l, l':R
H:eps > 0
0 <= Rabs l + Rabs l'
eps, l, l':R
H:eps > 0
(1 + 0 <= 1 + (Rabs l + Rabs l')) =
(1 <= 1 + (Rabs l + Rabs l'))
  cut (Rabs (l + l') <= Rabs l + Rabs l').
eps, l, l':R
H:eps > 0
Rabs (l + l') <= Rabs l + Rabs l' ->
0 <= Rabs l + Rabs l'
eps, l, l':R
H:eps > 0
Rabs (l + l') <= Rabs l + Rabs l'
eps, l, l':R
H:eps > 0
(1 + 0 <= 1 + (Rabs l + Rabs l')) =
(1 <= 1 + (Rabs l + Rabs l'))
  cut (0 <= Rabs (l + l')).
eps, l, l':R
H:eps > 0
0 <= Rabs (l + l') ->
Rabs (l + l') <= Rabs l + Rabs l' ->
0 <= Rabs l + Rabs l'
eps, l, l':R
H:eps > 0
0 <= Rabs (l + l')
eps, l, l':R
H:eps > 0
Rabs (l + l') <= Rabs l + Rabs l'
eps, l, l':R
H:eps > 0
(1 + 0 <= 1 + (Rabs l + Rabs l')) =
(1 <= 1 + (Rabs l + Rabs l'))
  exact (Rle_trans _ _ _).
eps, l, l':R
H:eps > 0
0 <= Rabs (l + l')
eps, l, l':R
H:eps > 0
Rabs (l + l') <= Rabs l + Rabs l'
eps, l, l':R
H:eps > 0
(1 + 0 <= 1 + (Rabs l + Rabs l')) =
(1 <= 1 + (Rabs l + Rabs l'))
  exact (Rabs_pos _).
eps, l, l':R
H:eps > 0
Rabs (l + l') <= Rabs l + Rabs l'
eps, l, l':R
H:eps > 0
(1 + 0 <= 1 + (Rabs l + Rabs l')) =
(1 <= 1 + (Rabs l + Rabs l'))
  exact (Rabs_triang _ _).
eps, l, l':R
H:eps > 0
(1 + 0 <= 1 + (Rabs l + Rabs l')) =
(1 <= 1 + (Rabs l + Rabs l'))
  rewrite (proj1 (Rplus_ne 1)); trivial.
Qed.

(*********)
Lemma mul_factor_gt_f :
  forall eps l l':R, eps > 0 -> Rmin 1 (eps * mul_factor l l') > 0.
forall eps l l' : R,
eps > 0 -> Rmin 1 (eps * mul_factor l l') > 0
  intros; apply Rmin_Rgt_r; split.
eps, l, l':R
H:eps > 0
1 > 0
eps, l, l':R
H:eps > 0
eps * mul_factor l l' > 0
  exact Rlt_0_1.
eps, l, l':R
H:eps > 0
eps * mul_factor l l' > 0
  exact (mul_factor_gt eps l l' H).
Qed.


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Record Metric_Space : Type :=
  {Base : Type;
    dist : Base -> Base -> R;
    dist_pos : forall x y:Base, dist x y >= 0;
    dist_sym : forall x y:Base, dist x y = dist y x;
    dist_refl : forall x y:Base, dist x y = 0 <-> x = y;
    dist_tri : forall x y z:Base, dist x y <= dist x z + dist z y}.

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(*********)
Definition limit_in (X X':Metric_Space) (f:Base X -> Base X')
  (D:Base X -> Prop) (x0:Base X) (l:Base X') :=
  forall eps:R,
    eps > 0 ->
    exists alp : R,
      alp > 0 /\
      (forall x:Base X, D x /\ (dist X) x x0 < alp -> (dist X') (f x) l < eps).

(*******************************)

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(*********)
Definition R_met : Metric_Space :=
  Build_Metric_Space R R_dist R_dist_pos R_dist_sym R_dist_refl R_dist_tri.

Declare Equivalent Keys dist R_dist.

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Definition Dgf (Df Dg:R -> Prop) (f:R -> R) (x:R) := Df x /\ Dg (f x).

(*********)
Definition limit1_in (f:R -> R) (D:R -> Prop) (l x0:R) : Prop :=
  limit_in R_met R_met f D x0 l.

(*********)
Lemma tech_limit :
  forall (f:R -> R) (D:R -> Prop) (l x0:R),
    D x0 -> limit1_in f D l x0 -> l = f x0.
forall (f : R -> R) (D : R -> Prop) (l x0 : R),
D x0 -> limit1_in f D l x0 -> l = f x0
Proof.
forall (f : R -> R) (D : R -> Prop) (l x0 : R),
D x0 -> limit1_in f D l x0 -> l = f x0
  intros f D l x0 H H0.
f:R -> R
D:R -> Prop
l, x0:R
H:D x0
H0:limit1_in f D l x0
l = f x0
  case (Rabs_pos (f x0 - l)); intros H1.
f:R -> R
D:R -> Prop
l, x0:R
H:D x0
H0:limit1_in f D l x0
H1:0 < Rabs (f x0 - l)
l = f x0
f:R -> R
D:R -> Prop
l, x0:R
H:D x0
H0:limit1_in f D l x0
H1:0 = Rabs (f x0 - l)
l = f x0
  absurd ((@dist R_met) (f x0) l < (@dist R_met) (f x0) l).
f:R -> R
D:R -> Prop
l, x0:R
H:D x0
H0:limit1_in f D l x0
H1:0 < Rabs (f x0 - l)
~ dist R_met (f x0) l < dist R_met (f x0) l
f:R -> R
D:R -> Prop
l, x0:R
H:D x0
H0:limit1_in f D l x0
H1:0 < Rabs (f x0 - l)
dist R_met (f x0) l < dist R_met (f x0) l
f:R -> R
D:R -> Prop
l, x0:R
H:D x0
H0:limit1_in f D l x0
H1:0 = Rabs (f x0 - l)
l = f x0
  apply Rlt_irrefl.
f:R -> R
D:R -> Prop
l, x0:R
H:D x0
H0:limit1_in f D l x0
H1:0 < Rabs (f x0 - l)
dist R_met (f x0) l < dist R_met (f x0) l
f:R -> R
D:R -> Prop
l, x0:R
H:D x0
H0:limit1_in f D l x0
H1:0 = Rabs (f x0 - l)
l = f x0
  case (H0 ((@dist R_met) (f x0) l)); auto.
f:R -> R
D:R -> Prop
l, x0:R
H:D x0
H0:limit1_in f D l x0
H1:0 < Rabs (f x0 - l)
forall x : R,
x > 0 /\
(forall x1 : Base R_met,
 D x1 /\ dist R_met x1 x0 < x ->
 dist R_met (f x1) l < dist R_met (f x0) l) ->
dist R_met (f x0) l < dist R_met (f x0) l
f:R -> R
D:R -> Prop
l, x0:R
H:D x0
H0:limit1_in f D l x0
H1:0 = Rabs (f x0 - l)
l = f x0
  intros alpha1 [H2 H3]; apply H3; auto; split; auto.
f:R -> R
D:R -> Prop
l, x0:R
H:D x0
H0:limit1_in f D l x0
H1:0 < Rabs (f x0 - l)
alpha1:R
H2:alpha1 > 0
H3:forall x : Base R_met,
D x /\ dist R_met x x0 < alpha1 ->
dist R_met (f x) l < dist R_met (f x0) l
dist R_met x0 x0 < alpha1
f:R -> R
D:R -> Prop
l, x0:R
H:D x0
H0:limit1_in f D l x0
H1:0 = Rabs (f x0 - l)
l = f x0
  case (dist_refl R_met x0 x0); intros Hr1 Hr2; rewrite Hr2; auto.
f:R -> R
D:R -> Prop
l, x0:R
H:D x0
H0:limit1_in f D l x0
H1:0 = Rabs (f x0 - l)
l = f x0
  case (dist_refl R_met (f x0) l); intros Hr1 Hr2; symmetry; auto.
Qed.

(*********)
Lemma tech_limit_contr :
  forall (f:R -> R) (D:R -> Prop) (l x0:R),
    D x0 -> l <> f x0 -> ~ limit1_in f D l x0.
forall (f : R -> R) (D : R -> Prop) (l x0 : R),
D x0 -> l <> f x0 -> ~ limit1_in f D l x0
Proof.
forall (f : R -> R) (D : R -> Prop) (l x0 : R),
D x0 -> l <> f x0 -> ~ limit1_in f D l x0
  intros; generalize (tech_limit f D l x0); tauto.
Qed.

(*********)
Lemma lim_x : forall (D:R -> Prop) (x0:R), limit1_in (fun x:R => x) D x0 x0.
forall (D : R -> Prop) (x0 : R),
limit1_in (fun x : R => x) D x0 x0
Proof.
forall (D : R -> Prop) (x0 : R),
limit1_in (fun x : R => x) D x0 x0
  unfold limit1_in; unfold limit_in; simpl; intros;
    split with eps; split; auto; intros; elim H0; intros;
      auto.
Qed.

(*********)
Lemma limit_plus :
  forall (f g:R -> R) (D:R -> Prop) (l l' x0:R),
    limit1_in f D l x0 ->
    limit1_in g D l' x0 -> limit1_in (fun x:R => f x + g x) D (l + l') x0.
forall (f g : R -> R) (D : R -> Prop) (l l' x0 : R),
limit1_in f D l x0 ->
limit1_in g D l' x0 ->
limit1_in (fun x : R => f x + g x) D (l + l') x0
Proof.
forall (f g : R -> R) (D : R -> Prop) (l l' x0 : R),
limit1_in f D l x0 ->
limit1_in g D l' x0 ->
limit1_in (fun x : R => f x + g x) D (l + l') x0
  intros; unfold limit1_in; unfold limit_in; simpl;
    intros; elim (H (eps * / 2) (eps2_Rgt_R0 eps H1));
      elim (H0 (eps * / 2) (eps2_Rgt_R0 eps H1)); simpl;
        clear H H0; intros; elim H; elim H0; clear H H0; intros;
          split with (Rmin x1 x); split.
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x2 : R,
D x2 /\ R_dist x2 x0 < x1 ->
R_dist (f x2) l < eps * / 2
H2:x > 0
H3:forall x2 : R,
D x2 /\ R_dist x2 x0 < x ->
R_dist (g x2) l' < eps * / 2
Rmin x1 x > 0
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x2 : R,
D x2 /\ R_dist x2 x0 < x1 ->
R_dist (f x2) l < eps * / 2
H2:x > 0
H3:forall x2 : R,
D x2 /\ R_dist x2 x0 < x ->
R_dist (g x2) l' < eps * / 2
forall x2 : R,
D x2 /\ R_dist x2 x0 < Rmin x1 x ->
R_dist (f x2 + g x2) (l + l') < eps
  exact (Rmin_Rgt_r x1 x 0 (conj H H2)).
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x2 : R,
D x2 /\ R_dist x2 x0 < x1 ->
R_dist (f x2) l < eps * / 2
H2:x > 0
H3:forall x2 : R,
D x2 /\ R_dist x2 x0 < x ->
R_dist (g x2) l' < eps * / 2
forall x2 : R,
D x2 /\ R_dist x2 x0 < Rmin x1 x ->
R_dist (f x2 + g x2) (l + l') < eps
  intros; elim H4; clear H4; intros;
    cut (R_dist (f x2) l + R_dist (g x2) l' < eps).
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < eps * / 2
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * / 2
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
R_dist (f x2) l + R_dist (g x2) l' < eps ->
R_dist (f x2 + g x2) (l + l') < eps
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < eps * / 2
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * / 2
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
R_dist (f x2) l + R_dist (g x2) l' < eps
  cut (R_dist (f x2 + g x2) (l + l') <= R_dist (f x2) l + R_dist (g x2) l').
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < eps * / 2
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * / 2
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
R_dist (f x2 + g x2) (l + l') <=
R_dist (f x2) l + R_dist (g x2) l' ->
R_dist (f x2) l + R_dist (g x2) l' < eps ->
R_dist (f x2 + g x2) (l + l') < eps
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < eps * / 2
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * / 2
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
R_dist (f x2 + g x2) (l + l') <=
R_dist (f x2) l + R_dist (g x2) l'
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < eps * / 2
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * / 2
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
R_dist (f x2) l + R_dist (g x2) l' < eps
  exact (Rle_lt_trans _ _ _).
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < eps * / 2
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * / 2
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
R_dist (f x2 + g x2) (l + l') <=
R_dist (f x2) l + R_dist (g x2) l'
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < eps * / 2
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * / 2
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
R_dist (f x2) l + R_dist (g x2) l' < eps
  exact (R_dist_plus _ _ _ _).
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < eps * / 2
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * / 2
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
R_dist (f x2) l + R_dist (g x2) l' < eps
  elim (Rmin_Rgt_l x1 x (R_dist x2 x0) H5); clear H5; intros.
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < eps * / 2
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * / 2
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
R_dist (f x2) l + R_dist (g x2) l' < eps
  generalize (H3 x2 (conj H4 H6)); generalize (H0 x2 (conj H4 H5)); intros;
    replace eps with (eps * / 2 + eps * / 2).
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < eps * / 2
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * / 2
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < eps * / 2
H8:R_dist (g x2) l' < eps * / 2
R_dist (f x2) l + R_dist (g x2) l' <
eps * / 2 + eps * / 2
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < eps * / 2
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * / 2
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < eps * / 2
H8:R_dist (g x2) l' < eps * / 2
eps * / 2 + eps * / 2 = eps
  exact (Rplus_lt_compat _ _ _ _ H7 H8).
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < eps * / 2
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * / 2
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < eps * / 2
H8:R_dist (g x2) l' < eps * / 2
eps * / 2 + eps * / 2 = eps
  exact (eps2 eps).
Qed.

(*********)
Lemma limit_Ropp :
  forall (f:R -> R) (D:R -> Prop) (l x0:R),
    limit1_in f D l x0 -> limit1_in (fun x:R => - f x) D (- l) x0.
forall (f : R -> R) (D : R -> Prop) (l x0 : R),
limit1_in f D l x0 ->
limit1_in (fun x : R => - f x) D (- l) x0
Proof.
forall (f : R -> R) (D : R -> Prop) (l x0 : R),
limit1_in f D l x0 ->
limit1_in (fun x : R => - f x) D (- l) x0
  unfold limit1_in; unfold limit_in; simpl; intros;
    elim (H eps H0); clear H; intros; elim H; clear H;
      intros; split with x; split; auto; intros; generalize (H1 x1 H2);
        clear H1; intro; unfold R_dist; unfold Rminus;
          rewrite (Ropp_involutive l); rewrite (Rplus_comm (- f x1) l);
            fold (l - f x1); fold (R_dist l (f x1));
              rewrite R_dist_sym; assumption.
Qed.

(*********)
Lemma limit_minus :
  forall (f g:R -> R) (D:R -> Prop) (l l' x0:R),
    limit1_in f D l x0 ->
    limit1_in g D l' x0 -> limit1_in (fun x:R => f x - g x) D (l - l') x0.
forall (f g : R -> R) (D : R -> Prop) (l l' x0 : R),
limit1_in f D l x0 ->
limit1_in g D l' x0 ->
limit1_in (fun x : R => f x - g x) D (l - l') x0
Proof.
forall (f g : R -> R) (D : R -> Prop) (l l' x0 : R),
limit1_in f D l x0 ->
limit1_in g D l' x0 ->
limit1_in (fun x : R => f x - g x) D (l - l') x0
  intros; unfold Rminus; generalize (limit_Ropp g D l' x0 H0); intro;
    exact (limit_plus f (fun x:R => - g x) D l (- l') x0 H H1).
Qed.

(*********)
Lemma limit_free :
  forall (f:R -> R) (D:R -> Prop) (x x0:R),
    limit1_in (fun h:R => f x) D (f x) x0.
forall (f : R -> R) (D : R -> Prop) (x x0 : R),
limit1_in (fun _ : R => f x) D (f x) x0
Proof.
forall (f : R -> R) (D : R -> Prop) (x x0 : R),
limit1_in (fun _ : R => f x) D (f x) x0
  unfold limit1_in; unfold limit_in; simpl; intros;
    split with eps; split; auto; intros; elim (R_dist_refl (f x) (f x));
      intros a b; rewrite (b (eq_refl (f x))); unfold Rgt in H;
        assumption.
Qed.

(*********)
Lemma limit_mul :
  forall (f g:R -> R) (D:R -> Prop) (l l' x0:R),
    limit1_in f D l x0 ->
    limit1_in g D l' x0 -> limit1_in (fun x:R => f x * g x) D (l * l') x0.
forall (f g : R -> R) (D : R -> Prop) (l l' x0 : R),
limit1_in f D l x0 ->
limit1_in g D l' x0 ->
limit1_in (fun x : R => f x * g x) D (l * l') x0
Proof.
forall (f g : R -> R) (D : R -> Prop) (l l' x0 : R),
limit1_in f D l x0 ->
limit1_in g D l' x0 ->
limit1_in (fun x : R => f x * g x) D (l * l') x0
  intros; unfold limit1_in; unfold limit_in; simpl;
    intros;
      elim (H (Rmin 1 (eps * mul_factor l l')) (mul_factor_gt_f eps l l' H1));
        elim (H0 (eps * mul_factor l l') (mul_factor_gt eps l l' H1));
          clear H H0; simpl; intros; elim H; elim H0;
            clear H H0; intros; split with (Rmin x1 x); split.
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x2 : R,
D x2 /\ R_dist x2 x0 < x1 ->
R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x2 : R,
D x2 /\ R_dist x2 x0 < x ->
R_dist (g x2) l' < eps * mul_factor l l'
Rmin x1 x > 0
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x2 : R,
D x2 /\ R_dist x2 x0 < x1 ->
R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x2 : R,
D x2 /\ R_dist x2 x0 < x ->
R_dist (g x2) l' < eps * mul_factor l l'
forall x2 : R,
D x2 /\ R_dist x2 x0 < Rmin x1 x ->
R_dist (f x2 * g x2) (l * l') < eps
  exact (Rmin_Rgt_r x1 x 0 (conj H H2)).
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x2 : R,
D x2 /\ R_dist x2 x0 < x1 ->
R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x2 : R,
D x2 /\ R_dist x2 x0 < x ->
R_dist (g x2) l' < eps * mul_factor l l'
forall x2 : R,
D x2 /\ R_dist x2 x0 < Rmin x1 x ->
R_dist (f x2 * g x2) (l * l') < eps
  intros; elim H4; clear H4; intros; unfold R_dist;
    replace (f x2 * g x2 - l * l') with (f x2 * (g x2 - l') + l' * (f x2 - l)).
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
Rabs (f x2 * (g x2 - l') + l' * (f x2 - l)) < eps
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
f x2 * (g x2 - l') + l' * (f x2 - l) =
f x2 * g x2 - l * l'
  cut (Rabs (f x2 * (g x2 - l')) + Rabs (l' * (f x2 - l)) < eps).
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
Rabs (f x2 * (g x2 - l')) + Rabs (l' * (f x2 - l)) <
eps ->
Rabs (f x2 * (g x2 - l') + l' * (f x2 - l)) < eps
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
Rabs (f x2 * (g x2 - l')) + Rabs (l' * (f x2 - l)) <
eps
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
f x2 * (g x2 - l') + l' * (f x2 - l) =
f x2 * g x2 - l * l'
  cut
    (Rabs (f x2 * (g x2 - l') + l' * (f x2 - l)) <=
      Rabs (f x2 * (g x2 - l')) + Rabs (l' * (f x2 - l))).
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
Rabs (f x2 * (g x2 - l') + l' * (f x2 - l)) <=
Rabs (f x2 * (g x2 - l')) + Rabs (l' * (f x2 - l)) ->
Rabs (f x2 * (g x2 - l')) + Rabs (l' * (f x2 - l)) <
eps ->
Rabs (f x2 * (g x2 - l') + l' * (f x2 - l)) < eps
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
Rabs (f x2 * (g x2 - l') + l' * (f x2 - l)) <=
Rabs (f x2 * (g x2 - l')) + Rabs (l' * (f x2 - l))
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
Rabs (f x2 * (g x2 - l')) + Rabs (l' * (f x2 - l)) <
eps
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
f x2 * (g x2 - l') + l' * (f x2 - l) =
f x2 * g x2 - l * l'
  exact (Rle_lt_trans _ _ _).
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
Rabs (f x2 * (g x2 - l') + l' * (f x2 - l)) <=
Rabs (f x2 * (g x2 - l')) + Rabs (l' * (f x2 - l))
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
Rabs (f x2 * (g x2 - l')) + Rabs (l' * (f x2 - l)) <
eps
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
f x2 * (g x2 - l') + l' * (f x2 - l) =
f x2 * g x2 - l * l'
  exact (Rabs_triang _ _).
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
Rabs (f x2 * (g x2 - l')) + Rabs (l' * (f x2 - l)) <
eps
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
f x2 * (g x2 - l') + l' * (f x2 - l) =
f x2 * g x2 - l * l'
  rewrite (Rabs_mult (f x2) (g x2 - l')); rewrite (Rabs_mult l' (f x2 - l));
    cut
      ((1 + Rabs l) * (eps * mul_factor l l') + Rabs l' * (eps * mul_factor l l') <=
        eps).
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
(1 + Rabs l) * (eps * mul_factor l l') +
Rabs l' * (eps * mul_factor l l') <= eps ->
Rabs (f x2) * Rabs (g x2 - l') +
Rabs l' * Rabs (f x2 - l) < eps
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
(1 + Rabs l) * (eps * mul_factor l l') +
Rabs l' * (eps * mul_factor l l') <= eps
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
f x2 * (g x2 - l') + l' * (f x2 - l) =
f x2 * g x2 - l * l'
  cut
    (Rabs (f x2) * Rabs (g x2 - l') + Rabs l' * Rabs (f x2 - l) <
      (1 + Rabs l) * (eps * mul_factor l l') + Rabs l' * (eps * mul_factor l l')).
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
Rabs (f x2) * Rabs (g x2 - l') +
Rabs l' * Rabs (f x2 - l) <
(1 + Rabs l) * (eps * mul_factor l l') +
Rabs l' * (eps * mul_factor l l') ->
(1 + Rabs l) * (eps * mul_factor l l') +
Rabs l' * (eps * mul_factor l l') <= eps ->
Rabs (f x2) * Rabs (g x2 - l') +
Rabs l' * Rabs (f x2 - l) < eps
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
Rabs (f x2) * Rabs (g x2 - l') +
Rabs l' * Rabs (f x2 - l) <
(1 + Rabs l) * (eps * mul_factor l l') +
Rabs l' * (eps * mul_factor l l')
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
(1 + Rabs l) * (eps * mul_factor l l') +
Rabs l' * (eps * mul_factor l l') <= eps
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
f x2 * (g x2 - l') + l' * (f x2 - l) =
f x2 * g x2 - l * l'
  exact (Rlt_le_trans _ _ _).
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
Rabs (f x2) * Rabs (g x2 - l') +
Rabs l' * Rabs (f x2 - l) <
(1 + Rabs l) * (eps * mul_factor l l') +
Rabs l' * (eps * mul_factor l l')
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
(1 + Rabs l) * (eps * mul_factor l l') +
Rabs l' * (eps * mul_factor l l') <= eps
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
f x2 * (g x2 - l') + l' * (f x2 - l) =
f x2 * g x2 - l * l'
  elim (Rmin_Rgt_l x1 x (R_dist x2 x0) H5); clear H5; intros;
    generalize (H0 x2 (conj H4 H5)); intro; generalize (Rmin_Rgt_l _ _ _ H7);
      intro; elim H8; intros; clear H0 H8; apply Rplus_lt_le_compat.
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > R_dist (f x2) l
H10:eps * mul_factor l l' > R_dist (f x2) l
Rabs (f x2) * Rabs (g x2 - l') <
(1 + Rabs l) * (eps * mul_factor l l')
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > R_dist (f x2) l
H10:eps * mul_factor l l' > R_dist (f x2) l
Rabs l' * Rabs (f x2 - l) <=
Rabs l' * (eps * mul_factor l l')
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
(1 + Rabs l) * (eps * mul_factor l l') +
Rabs l' * (eps * mul_factor l l') <= eps
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
f x2 * (g x2 - l') + l' * (f x2 - l) =
f x2 * g x2 - l * l'
  apply Rmult_ge_0_gt_0_lt_compat.
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > R_dist (f x2) l
H10:eps * mul_factor l l' > R_dist (f x2) l
Rabs (g x2 - l') >= 0
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > R_dist (f x2) l
H10:eps * mul_factor l l' > R_dist (f x2) l
1 + Rabs l > 0
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > R_dist (f x2) l
H10:eps * mul_factor l l' > R_dist (f x2) l
Rabs (f x2) < 1 + Rabs l
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > R_dist (f x2) l
H10:eps * mul_factor l l' > R_dist (f x2) l
Rabs (g x2 - l') < eps * mul_factor l l'
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > R_dist (f x2) l
H10:eps * mul_factor l l' > R_dist (f x2) l
Rabs l' * Rabs (f x2 - l) <=
Rabs l' * (eps * mul_factor l l')
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
(1 + Rabs l) * (eps * mul_factor l l') +
Rabs l' * (eps * mul_factor l l') <= eps
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
f x2 * (g x2 - l') + l' * (f x2 - l) =
f x2 * g x2 - l * l'
  apply Rle_ge.
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > R_dist (f x2) l
H10:eps * mul_factor l l' > R_dist (f x2) l
0 <= Rabs (g x2 - l')
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > R_dist (f x2) l
H10:eps * mul_factor l l' > R_dist (f x2) l
1 + Rabs l > 0
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > R_dist (f x2) l
H10:eps * mul_factor l l' > R_dist (f x2) l
Rabs (f x2) < 1 + Rabs l
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > R_dist (f x2) l
H10:eps * mul_factor l l' > R_dist (f x2) l
Rabs (g x2 - l') < eps * mul_factor l l'
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > R_dist (f x2) l
H10:eps * mul_factor l l' > R_dist (f x2) l
Rabs l' * Rabs (f x2 - l) <=
Rabs l' * (eps * mul_factor l l')
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
(1 + Rabs l) * (eps * mul_factor l l') +
Rabs l' * (eps * mul_factor l l') <= eps
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
f x2 * (g x2 - l') + l' * (f x2 - l) =
f x2 * g x2 - l * l'
  exact (Rabs_pos (g x2 - l')).
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > R_dist (f x2) l
H10:eps * mul_factor l l' > R_dist (f x2) l
1 + Rabs l > 0
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > R_dist (f x2) l
H10:eps * mul_factor l l' > R_dist (f x2) l
Rabs (f x2) < 1 + Rabs l
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > R_dist (f x2) l
H10:eps * mul_factor l l' > R_dist (f x2) l
Rabs (g x2 - l') < eps * mul_factor l l'
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > R_dist (f x2) l
H10:eps * mul_factor l l' > R_dist (f x2) l
Rabs l' * Rabs (f x2 - l) <=
Rabs l' * (eps * mul_factor l l')
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
(1 + Rabs l) * (eps * mul_factor l l') +
Rabs l' * (eps * mul_factor l l') <= eps
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
f x2 * (g x2 - l') + l' * (f x2 - l) =
f x2 * g x2 - l * l'
  rewrite (Rplus_comm 1 (Rabs l)); unfold Rgt; apply Rle_lt_0_plus_1;
    exact (Rabs_pos l).
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > R_dist (f x2) l
H10:eps * mul_factor l l' > R_dist (f x2) l
Rabs (f x2) < 1 + Rabs l
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > R_dist (f x2) l
H10:eps * mul_factor l l' > R_dist (f x2) l
Rabs (g x2 - l') < eps * mul_factor l l'
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > R_dist (f x2) l
H10:eps * mul_factor l l' > R_dist (f x2) l
Rabs l' * Rabs (f x2 - l) <=
Rabs l' * (eps * mul_factor l l')
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
(1 + Rabs l) * (eps * mul_factor l l') +
Rabs l' * (eps * mul_factor l l') <= eps
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
f x2 * (g x2 - l') + l' * (f x2 - l) =
f x2 * g x2 - l * l'
  unfold R_dist in H9;
    apply (Rplus_lt_reg_l (- Rabs l) (Rabs (f x2)) (1 + Rabs l)).
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > Rabs (f x2 - l)
H10:eps * mul_factor l l' > R_dist (f x2) l
- Rabs l + Rabs (f x2) < - Rabs l + (1 + Rabs l)
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > R_dist (f x2) l
H10:eps * mul_factor l l' > R_dist (f x2) l
Rabs (g x2 - l') < eps * mul_factor l l'
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > R_dist (f x2) l
H10:eps * mul_factor l l' > R_dist (f x2) l
Rabs l' * Rabs (f x2 - l) <=
Rabs l' * (eps * mul_factor l l')
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
(1 + Rabs l) * (eps * mul_factor l l') +
Rabs l' * (eps * mul_factor l l') <= eps
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
f x2 * (g x2 - l') + l' * (f x2 - l) =
f x2 * g x2 - l * l'
  rewrite <- (Rplus_assoc (- Rabs l) 1 (Rabs l));
    rewrite (Rplus_comm (- Rabs l) 1);
      rewrite (Rplus_assoc 1 (- Rabs l) (Rabs l)); rewrite (Rplus_opp_l (Rabs l));
        rewrite (proj1 (Rplus_ne 1)); rewrite (Rplus_comm (- Rabs l) (Rabs (f x2)));
          generalize H9; cut (Rabs (f x2) - Rabs l <= Rabs (f x2 - l)).
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > Rabs (f x2 - l)
H10:eps * mul_factor l l' > R_dist (f x2) l
Rabs (f x2) - Rabs l <= Rabs (f x2 - l) ->
1 > Rabs (f x2 - l) -> Rabs (f x2) + - Rabs l < 1
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > Rabs (f x2 - l)
H10:eps * mul_factor l l' > R_dist (f x2) l
Rabs (f x2) - Rabs l <= Rabs (f x2 - l)
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > R_dist (f x2) l
H10:eps * mul_factor l l' > R_dist (f x2) l
Rabs (g x2 - l') < eps * mul_factor l l'
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > R_dist (f x2) l
H10:eps * mul_factor l l' > R_dist (f x2) l
Rabs l' * Rabs (f x2 - l) <=
Rabs l' * (eps * mul_factor l l')
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
(1 + Rabs l) * (eps * mul_factor l l') +
Rabs l' * (eps * mul_factor l l') <= eps
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
f x2 * (g x2 - l') + l' * (f x2 - l) =
f x2 * g x2 - l * l'
  exact (Rle_lt_trans _ _ _).
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > Rabs (f x2 - l)
H10:eps * mul_factor l l' > R_dist (f x2) l
Rabs (f x2) - Rabs l <= Rabs (f x2 - l)
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > R_dist (f x2) l
H10:eps * mul_factor l l' > R_dist (f x2) l
Rabs (g x2 - l') < eps * mul_factor l l'
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > R_dist (f x2) l
H10:eps * mul_factor l l' > R_dist (f x2) l
Rabs l' * Rabs (f x2 - l) <=
Rabs l' * (eps * mul_factor l l')
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
(1 + Rabs l) * (eps * mul_factor l l') +
Rabs l' * (eps * mul_factor l l') <= eps
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
f x2 * (g x2 - l') + l' * (f x2 - l) =
f x2 * g x2 - l * l'
  exact (Rabs_triang_inv _ _).
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > R_dist (f x2) l
H10:eps * mul_factor l l' > R_dist (f x2) l
Rabs (g x2 - l') < eps * mul_factor l l'
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > R_dist (f x2) l
H10:eps * mul_factor l l' > R_dist (f x2) l
Rabs l' * Rabs (f x2 - l) <=
Rabs l' * (eps * mul_factor l l')
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
(1 + Rabs l) * (eps * mul_factor l l') +
Rabs l' * (eps * mul_factor l l') <= eps
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
f x2 * (g x2 - l') + l' * (f x2 - l) =
f x2 * g x2 - l * l'
  generalize (H3 x2 (conj H4 H6)); trivial.
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > R_dist (f x2) l
H10:eps * mul_factor l l' > R_dist (f x2) l
Rabs l' * Rabs (f x2 - l) <=
Rabs l' * (eps * mul_factor l l')
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
(1 + Rabs l) * (eps * mul_factor l l') +
Rabs l' * (eps * mul_factor l l') <= eps
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
f x2 * (g x2 - l') + l' * (f x2 - l) =
f x2 * g x2 - l * l'
  apply Rmult_le_compat_l.
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > R_dist (f x2) l
H10:eps * mul_factor l l' > R_dist (f x2) l
0 <= Rabs l'
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > R_dist (f x2) l
H10:eps * mul_factor l l' > R_dist (f x2) l
Rabs (f x2 - l) <= eps * mul_factor l l'
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
(1 + Rabs l) * (eps * mul_factor l l') +
Rabs l' * (eps * mul_factor l l') <= eps
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
f x2 * (g x2 - l') + l' * (f x2 - l) =
f x2 * g x2 - l * l'
  exact (Rabs_pos l').
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:x1 > R_dist x2 x0
H6:x > R_dist x2 x0
H7:R_dist (f x2) l < Rmin 1 (eps * mul_factor l l')
H9:1 > R_dist (f x2) l
H10:eps * mul_factor l l' > R_dist (f x2) l
Rabs (f x2 - l) <= eps * mul_factor l l'
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
(1 + Rabs l) * (eps * mul_factor l l') +
Rabs l' * (eps * mul_factor l l') <= eps
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
f x2 * (g x2 - l') + l' * (f x2 - l) =
f x2 * g x2 - l * l'
  unfold Rle; left; assumption.
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
(1 + Rabs l) * (eps * mul_factor l l') +
Rabs l' * (eps * mul_factor l l') <= eps
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
f x2 * (g x2 - l') + l' * (f x2 - l) =
f x2 * g x2 - l * l'
  rewrite (Rmult_comm (1 + Rabs l) (eps * mul_factor l l'));
    rewrite (Rmult_comm (Rabs l') (eps * mul_factor l l'));
      rewrite <-
        (Rmult_plus_distr_l (eps * mul_factor l l') (1 + Rabs l) (Rabs l'))
        ; rewrite (Rmult_assoc eps (mul_factor l l') (1 + Rabs l + Rabs l'));
          rewrite (Rplus_assoc 1 (Rabs l) (Rabs l')); unfold mul_factor;
            rewrite (Rinv_l (1 + (Rabs l + Rabs l')) (mul_factor_wd l l'));
              rewrite (proj1 (Rmult_ne eps)); apply Req_le; trivial.
f, g:R -> R
D:R -> Prop
l, l', x0, eps:R
H1:eps > 0
x, x1:R
H:x1 > 0
H0:forall x3 : R,
D x3 /\ R_dist x3 x0 < x1 ->
R_dist (f x3) l < Rmin 1 (eps * mul_factor l l')
H2:x > 0
H3:forall x3 : R,
D x3 /\ R_dist x3 x0 < x ->
R_dist (g x3) l' < eps * mul_factor l l'
x2:R
H4:D x2
H5:R_dist x2 x0 < Rmin x1 x
f x2 * (g x2 - l') + l' * (f x2 - l) =
f x2 * g x2 - l * l'
  ring.
Qed.

(*********)
Definition adhDa (D:R -> Prop) (a:R) : Prop :=
  forall alp:R, alp > 0 ->  exists x : R, D x /\ R_dist x a < alp.

(*********)
Lemma single_limit :
  forall (f:R -> R) (D:R -> Prop) (l l' x0:R),
    adhDa D x0 -> limit1_in f D l x0 -> limit1_in f D l' x0 -> l = l'.
forall (f : R -> R) (D : R -> Prop) (l l' x0 : R),
adhDa D x0 ->
limit1_in f D l x0 -> limit1_in f D l' x0 -> l = l'
Proof.
forall (f : R -> R) (D : R -> Prop) (l l' x0 : R),
adhDa D x0 ->
limit1_in f D l x0 -> limit1_in f D l' x0 -> l = l'
  unfold limit1_in; unfold limit_in; intros.
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
H0:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D x /\ dist R_met x x0 < alp ->
   dist R_met (f x) l < eps)
H1:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : Base R_met,
   D x /\ dist R_met x x0 < alp ->
   dist R_met (f x) l' < eps)
l = l'
  simpl in *.
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
H0:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps)
H1:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l' < eps)
l = l'
  cut (forall eps:R, eps > 0 -> dist R_met l l' < 2 * eps).
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
H0:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps)
H1:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l' < eps)
(forall eps : R, eps > 0 -> dist R_met l l' < 2 * eps) ->
l = l'
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
H0:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps)
H1:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l' < eps)
forall eps : R, eps > 0 -> dist R_met l l' < 2 * eps
  clear H0 H1; unfold dist in |- *; unfold R_met; unfold R_dist in |- *;
    unfold Rabs; case (Rcase_abs (l - l')) as [Hlt|Hge]; intros.
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hlt:l - l' < 0
H0:forall eps : R, eps > 0 -> - (l - l') < 2 * eps
l = l'
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hge:l - l' >= 0
H0:forall eps : R, eps > 0 -> l - l' < 2 * eps
l = l'
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
H0:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps)
H1:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l' < eps)
forall eps : R, eps > 0 -> dist R_met l l' < 2 * eps
  cut (forall eps:R, eps > 0 -> - (l - l') < eps).
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hlt:l - l' < 0
H0:forall eps : R, eps > 0 -> - (l - l') < 2 * eps
(forall eps : R, eps > 0 -> - (l - l') < eps) ->
l = l'
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hlt:l - l' < 0
H0:forall eps : R, eps > 0 -> - (l - l') < 2 * eps
forall eps : R, eps > 0 -> - (l - l') < eps
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hge:l - l' >= 0
H0:forall eps : R, eps > 0 -> l - l' < 2 * eps
l = l'
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
H0:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps)
H1:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l' < eps)
forall eps : R, eps > 0 -> dist R_met l l' < 2 * eps
  intro; generalize (prop_eps (- (l - l')) H1); intro;
    generalize (Ropp_gt_lt_0_contravar (l - l') Hlt); intro;
      unfold Rgt in H3; generalize (Rgt_not_le (- (l - l')) 0 H3);
        intro; exfalso; auto.
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hlt:l - l' < 0
H0:forall eps : R, eps > 0 -> - (l - l') < 2 * eps
forall eps : R, eps > 0 -> - (l - l') < eps
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hge:l - l' >= 0
H0:forall eps : R, eps > 0 -> l - l' < 2 * eps
l = l'
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
H0:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps)
H1:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l' < eps)
forall eps : R, eps > 0 -> dist R_met l l' < 2 * eps
  intros; cut (eps * / 2 > 0).
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hlt:l - l' < 0
H0:forall eps0 : R,
eps0 > 0 -> - (l - l') < 2 * eps0
eps:R
H1:eps > 0
eps * / 2 > 0 -> - (l - l') < eps
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hlt:l - l' < 0
H0:forall eps0 : R,
eps0 > 0 -> - (l - l') < 2 * eps0
eps:R
H1:eps > 0
eps * / 2 > 0
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hge:l - l' >= 0
H0:forall eps : R, eps > 0 -> l - l' < 2 * eps
l = l'
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
H0:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps)
H1:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l' < eps)
forall eps : R, eps > 0 -> dist R_met l l' < 2 * eps
  intro; generalize (H0 (eps * / 2) H2); rewrite (Rmult_comm eps (/ 2));
    rewrite <- (Rmult_assoc 2 (/ 2) eps); rewrite (Rinv_r 2).
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hlt:l - l' < 0
H0:forall eps0 : R,
eps0 > 0 -> - (l - l') < 2 * eps0
eps:R
H1:eps > 0
H2:eps * / 2 > 0
- (l - l') < 1 * eps -> - (l - l') < eps
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hlt:l - l' < 0
H0:forall eps0 : R,
eps0 > 0 -> - (l - l') < 2 * eps0
eps:R
H1:eps > 0
H2:eps * / 2 > 0
2 <> 0
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hlt:l - l' < 0
H0:forall eps0 : R,
eps0 > 0 -> - (l - l') < 2 * eps0
eps:R
H1:eps > 0
eps * / 2 > 0
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hge:l - l' >= 0
H0:forall eps : R, eps > 0 -> l - l' < 2 * eps
l = l'
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
H0:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps)
H1:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l' < eps)
forall eps : R, eps > 0 -> dist R_met l l' < 2 * eps
  elim (Rmult_ne eps); intros a b; rewrite b; clear a b; trivial.
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hlt:l - l' < 0
H0:forall eps0 : R,
eps0 > 0 -> - (l - l') < 2 * eps0
eps:R
H1:eps > 0
H2:eps * / 2 > 0
2 <> 0
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hlt:l - l' < 0
H0:forall eps0 : R,
eps0 > 0 -> - (l - l') < 2 * eps0
eps:R
H1:eps > 0
eps * / 2 > 0
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hge:l - l' >= 0
H0:forall eps : R, eps > 0 -> l - l' < 2 * eps
l = l'
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
H0:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps)
H1:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l' < eps)
forall eps : R, eps > 0 -> dist R_met l l' < 2 * eps 
  apply (Rlt_dichotomy_converse 2 0); right; generalize Rlt_0_1; intro;
    unfold Rgt; generalize (Rplus_lt_compat_l 1 0 1 H3);
      intro; elim (Rplus_ne 1); intros a b; rewrite a in H4;
        clear a b; apply (Rlt_trans 0 1 2 H3 H4).
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hlt:l - l' < 0
H0:forall eps0 : R,
eps0 > 0 -> - (l - l') < 2 * eps0
eps:R
H1:eps > 0
eps * / 2 > 0
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hge:l - l' >= 0
H0:forall eps : R, eps > 0 -> l - l' < 2 * eps
l = l'
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
H0:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps)
H1:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l' < eps)
forall eps : R, eps > 0 -> dist R_met l l' < 2 * eps
  unfold Rgt; unfold Rgt in H1; rewrite (Rmult_comm eps (/ 2));
    rewrite <- (Rmult_0_r (/ 2)); apply (Rmult_lt_compat_l (/ 2) 0 eps);
      auto.
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hlt:l - l' < 0
H0:forall eps0 : R,
eps0 > 0 -> - (l - l') < 2 * eps0
eps:R
H1:0 < eps
0 < / 2
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hge:l - l' >= 0
H0:forall eps : R, eps > 0 -> l - l' < 2 * eps
l = l'
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
H0:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps)
H1:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l' < eps)
forall eps : R, eps > 0 -> dist R_met l l' < 2 * eps
  apply (Rinv_0_lt_compat 2); cut (1 < 2).
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hlt:l - l' < 0
H0:forall eps0 : R,
eps0 > 0 -> - (l - l') < 2 * eps0
eps:R
H1:0 < eps
1 < 2 -> 0 < 2
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hlt:l - l' < 0
H0:forall eps0 : R,
eps0 > 0 -> - (l - l') < 2 * eps0
eps:R
H1:0 < eps
1 < 2
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hge:l - l' >= 0
H0:forall eps : R, eps > 0 -> l - l' < 2 * eps
l = l'
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
H0:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps)
H1:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l' < eps)
forall eps : R, eps > 0 -> dist R_met l l' < 2 * eps
  intro; apply (Rlt_trans 0 1 2 Rlt_0_1 H2).
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hlt:l - l' < 0
H0:forall eps0 : R,
eps0 > 0 -> - (l - l') < 2 * eps0
eps:R
H1:0 < eps
1 < 2
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hge:l - l' >= 0
H0:forall eps : R, eps > 0 -> l - l' < 2 * eps
l = l'
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
H0:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps)
H1:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l' < eps)
forall eps : R, eps > 0 -> dist R_met l l' < 2 * eps
  generalize (Rplus_lt_compat_l 1 0 1 Rlt_0_1); elim (Rplus_ne 1); intros a b;
    rewrite a; clear a b; trivial.
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hge:l - l' >= 0
H0:forall eps : R, eps > 0 -> l - l' < 2 * eps
l = l'
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
H0:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps)
H1:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l' < eps)
forall eps : R, eps > 0 -> dist R_met l l' < 2 * eps
(**)
  cut (forall eps:R, eps > 0 -> l - l' < eps).
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hge:l - l' >= 0
H0:forall eps : R, eps > 0 -> l - l' < 2 * eps
(forall eps : R, eps > 0 -> l - l' < eps) -> l = l'
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hge:l - l' >= 0
H0:forall eps : R, eps > 0 -> l - l' < 2 * eps
forall eps : R, eps > 0 -> l - l' < eps
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
H0:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps)
H1:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l' < eps)
forall eps : R, eps > 0 -> dist R_met l l' < 2 * eps
  intro; generalize (prop_eps (l - l') H1); intro; elim (Rle_le_eq (l - l') 0);
    intros a b; clear b; apply (Rminus_diag_uniq l l');
      apply a; split.
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hge:l - l' >= 0
H0:forall eps : R, eps > 0 -> l - l' < 2 * eps
H1:forall eps : R, eps > 0 -> l - l' < eps
H2:l - l' <= 0
a:l - l' <= 0 <= l - l' -> l - l' = 0
l - l' <= 0
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hge:l - l' >= 0
H0:forall eps : R, eps > 0 -> l - l' < 2 * eps
H1:forall eps : R, eps > 0 -> l - l' < eps
H2:l - l' <= 0
a:l - l' <= 0 <= l - l' -> l - l' = 0
0 <= l - l'
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hge:l - l' >= 0
H0:forall eps : R, eps > 0 -> l - l' < 2 * eps
forall eps : R, eps > 0 -> l - l' < eps
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
H0:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps)
H1:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l' < eps)
forall eps : R, eps > 0 -> dist R_met l l' < 2 * eps
  assumption.
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hge:l - l' >= 0
H0:forall eps : R, eps > 0 -> l - l' < 2 * eps
H1:forall eps : R, eps > 0 -> l - l' < eps
H2:l - l' <= 0
a:l - l' <= 0 <= l - l' -> l - l' = 0
0 <= l - l'
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hge:l - l' >= 0
H0:forall eps : R, eps > 0 -> l - l' < 2 * eps
forall eps : R, eps > 0 -> l - l' < eps
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
H0:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps)
H1:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l' < eps)
forall eps : R, eps > 0 -> dist R_met l l' < 2 * eps
  apply (Rge_le (l - l') 0 Hge).
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hge:l - l' >= 0
H0:forall eps : R, eps > 0 -> l - l' < 2 * eps
forall eps : R, eps > 0 -> l - l' < eps
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
H0:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps)
H1:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l' < eps)
forall eps : R, eps > 0 -> dist R_met l l' < 2 * eps
  intros; cut (eps * / 2 > 0).
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hge:l - l' >= 0
H0:forall eps0 : R, eps0 > 0 -> l - l' < 2 * eps0
eps:R
H1:eps > 0
eps * / 2 > 0 -> l - l' < eps
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hge:l - l' >= 0
H0:forall eps0 : R, eps0 > 0 -> l - l' < 2 * eps0
eps:R
H1:eps > 0
eps * / 2 > 0
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
H0:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps)
H1:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l' < eps)
forall eps : R, eps > 0 -> dist R_met l l' < 2 * eps
  intro; generalize (H0 (eps * / 2) H2); rewrite (Rmult_comm eps (/ 2));
    rewrite <- (Rmult_assoc 2 (/ 2) eps); rewrite (Rinv_r 2).
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hge:l - l' >= 0
H0:forall eps0 : R, eps0 > 0 -> l - l' < 2 * eps0
eps:R
H1:eps > 0
H2:eps * / 2 > 0
l - l' < 1 * eps -> l - l' < eps
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hge:l - l' >= 0
H0:forall eps0 : R, eps0 > 0 -> l - l' < 2 * eps0
eps:R
H1:eps > 0
H2:eps * / 2 > 0
2 <> 0
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hge:l - l' >= 0
H0:forall eps0 : R, eps0 > 0 -> l - l' < 2 * eps0
eps:R
H1:eps > 0
eps * / 2 > 0
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
H0:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps)
H1:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l' < eps)
forall eps : R, eps > 0 -> dist R_met l l' < 2 * eps
  elim (Rmult_ne eps); intros a b; rewrite b; clear a b; trivial.
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hge:l - l' >= 0
H0:forall eps0 : R, eps0 > 0 -> l - l' < 2 * eps0
eps:R
H1:eps > 0
H2:eps * / 2 > 0
2 <> 0
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hge:l - l' >= 0
H0:forall eps0 : R, eps0 > 0 -> l - l' < 2 * eps0
eps:R
H1:eps > 0
eps * / 2 > 0
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
H0:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps)
H1:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l' < eps)
forall eps : R, eps > 0 -> dist R_met l l' < 2 * eps
  apply (Rlt_dichotomy_converse 2 0); right; generalize Rlt_0_1; intro;
    unfold Rgt; generalize (Rplus_lt_compat_l 1 0 1 H3);
      intro; elim (Rplus_ne 1); intros a b; rewrite a in H4;
        clear a b; apply (Rlt_trans 0 1 2 H3 H4).
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hge:l - l' >= 0
H0:forall eps0 : R, eps0 > 0 -> l - l' < 2 * eps0
eps:R
H1:eps > 0
eps * / 2 > 0
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
H0:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps)
H1:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l' < eps)
forall eps : R, eps > 0 -> dist R_met l l' < 2 * eps
  unfold Rgt; unfold Rgt in H1; rewrite (Rmult_comm eps (/ 2));
    rewrite <- (Rmult_0_r (/ 2)); apply (Rmult_lt_compat_l (/ 2) 0 eps);
      auto.
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hge:l - l' >= 0
H0:forall eps0 : R, eps0 > 0 -> l - l' < 2 * eps0
eps:R
H1:0 < eps
0 < / 2
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
H0:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps)
H1:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l' < eps)
forall eps : R, eps > 0 -> dist R_met l l' < 2 * eps
  apply (Rinv_0_lt_compat 2); cut (1 < 2).
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hge:l - l' >= 0
H0:forall eps0 : R, eps0 > 0 -> l - l' < 2 * eps0
eps:R
H1:0 < eps
1 < 2 -> 0 < 2
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hge:l - l' >= 0
H0:forall eps0 : R, eps0 > 0 -> l - l' < 2 * eps0
eps:R
H1:0 < eps
1 < 2
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
H0:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps)
H1:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l' < eps)
forall eps : R, eps > 0 -> dist R_met l l' < 2 * eps
  intro; apply (Rlt_trans 0 1 2 Rlt_0_1 H2).
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
Hge:l - l' >= 0
H0:forall eps0 : R, eps0 > 0 -> l - l' < 2 * eps0
eps:R
H1:0 < eps
1 < 2
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
H0:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps)
H1:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l' < eps)
forall eps : R, eps > 0 -> dist R_met l l' < 2 * eps
  generalize (Rplus_lt_compat_l 1 0 1 Rlt_0_1); elim (Rplus_ne 1); intros a b;
    rewrite a; clear a b; trivial.
f:R -> R
D:R -> Prop
l, l', x0:R
H:adhDa D x0
H0:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps)
H1:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ R_dist x x0 < alp ->
   R_dist (f x) l' < eps)
forall eps : R, eps > 0 -> dist R_met l l' < 2 * eps
(**)
  intros; unfold adhDa in H; elim (H0 eps H2); intros; elim (H1 eps H2); intros;
    clear H0 H1; elim H3; elim H4; clear H3 H4; intros;
      simpl; simpl in H1, H4; generalize (Rmin_Rgt x x1 0);
        intro; elim H5; intros; clear H5; elim (H (Rmin x x1) (H7 (conj H3 H0)));
          intros; elim H5; intros; clear H5 H H6 H7;
            generalize (Rmin_Rgt x x1 (R_dist x2 x0)); intro;
              elim H; intros; clear H H6; unfold Rgt in H5; elim (H5 H9);
                intros; clear H5 H9; generalize (H1 x2 (conj H8 H6));
                  generalize (H4 x2 (conj H8 H)); clear H8 H H6 H1 H4 H0 H3;
                    intros;
                      generalize
                        (Rplus_lt_compat (R_dist (f x2) l) eps (R_dist (f x2) l') eps H H0);
                        unfold R_dist; intros; rewrite (Rabs_minus_sym (f x2) l) in H1;
                          rewrite (Rmult_comm 2 eps); replace (eps *2) with (eps + eps) by ring;
                              generalize (R_dist_tri l l' (f x2)); unfold R_dist;
                                intros;
                                  apply
                                    (Rle_lt_trans (Rabs (l - l')) (Rabs (l - f x2) + Rabs (f x2 - l'))
                                      (eps + eps) H3 H1).
Qed.

(*********)
Lemma limit_comp :
  forall (f g:R -> R) (Df Dg:R -> Prop) (l l' x0:R),
    limit1_in f Df l x0 ->
    limit1_in g Dg l' l -> limit1_in (fun x:R => g (f x)) (Dgf Df Dg f) l' x0.
forall (f g : R -> R) (Df Dg : R -> Prop)
  (l l' x0 : R),
limit1_in f Df l x0 ->
limit1_in g Dg l' l ->
limit1_in (fun x : R => g (f x)) (Dgf Df Dg f) l' x0
Proof.
forall (f g : R -> R) (Df Dg : R -> Prop)
  (l l' x0 : R),
limit1_in f Df l x0 ->
limit1_in g Dg l' l ->
limit1_in (fun x : R => g (f x)) (Dgf Df Dg f) l' x0
  unfold limit1_in, limit_in, Dgf; simpl.
forall (f g : R -> R) (Df Dg : R -> Prop)
  (l l' x0 : R),
(forall eps : R,
 eps > 0 ->
 exists alp : R,
   alp > 0 /\
   (forall x : R,
    Df x /\ R_dist x x0 < alp -> R_dist (f x) l < eps)) ->
(forall eps : R,
 eps > 0 ->
 exists alp : R,
   alp > 0 /\
   (forall x : R,
    Dg x /\ R_dist x l < alp -> R_dist (g x) l' < eps)) ->
forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   (Df x /\ Dg (f x)) /\ R_dist x x0 < alp ->
   R_dist (g (f x)) l' < eps)
  intros f g Df Dg l l' x0 Hf Hg eps eps_pos.
f, g:R -> R
Df, Dg:R -> Prop
l, l', x0:R
Hf:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   Df x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps0)
Hg:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   Dg x /\ R_dist x l < alp ->
   R_dist (g x) l' < eps0)
eps:R
eps_pos:eps > 0
exists alp : R,
  alp > 0 /\
  (forall x : R,
   (Df x /\ Dg (f x)) /\ R_dist x x0 < alp ->
   R_dist (g (f x)) l' < eps)
  elim (Hg eps eps_pos).
f, g:R -> R
Df, Dg:R -> Prop
l, l', x0:R
Hf:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   Df x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps0)
Hg:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   Dg x /\ R_dist x l < alp ->
   R_dist (g x) l' < eps0)
eps:R
eps_pos:eps > 0
forall x : R,
x > 0 /\
(forall x1 : R,
 Dg x1 /\ R_dist x1 l < x -> R_dist (g x1) l' < eps) ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   (Df x1 /\ Dg (f x1)) /\ R_dist x1 x0 < alp ->
   R_dist (g (f x1)) l' < eps)
  intros alpg lg.
f, g:R -> R
Df, Dg:R -> Prop
l, l', x0:R
Hf:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   Df x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps0)
Hg:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   Dg x /\ R_dist x l < alp ->
   R_dist (g x) l' < eps0)
eps:R
eps_pos:eps > 0
alpg:R
lg:alpg > 0 /\
(forall x : R,
 Dg x /\ R_dist x l < alpg ->
 R_dist (g x) l' < eps)
exists alp : R,
  alp > 0 /\
  (forall x : R,
   (Df x /\ Dg (f x)) /\ R_dist x x0 < alp ->
   R_dist (g (f x)) l' < eps)
  elim (Hf alpg).
f, g:R -> R
Df, Dg:R -> Prop
l, l', x0:R
Hf:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   Df x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps0)
Hg:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   Dg x /\ R_dist x l < alp ->
   R_dist (g x) l' < eps0)
eps:R
eps_pos:eps > 0
alpg:R
lg:alpg > 0 /\
(forall x : R,
 Dg x /\ R_dist x l < alpg ->
 R_dist (g x) l' < eps)
forall x : R,
x > 0 /\
(forall x1 : R,
 Df x1 /\ R_dist x1 x0 < x -> R_dist (f x1) l < alpg) ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   (Df x1 /\ Dg (f x1)) /\ R_dist x1 x0 < alp ->
   R_dist (g (f x1)) l' < eps)
f, g:R -> R
Df, Dg:R -> Prop
l, l', x0:R
Hf:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   Df x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps0)
Hg:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   Dg x /\ R_dist x l < alp ->
   R_dist (g x) l' < eps0)
eps:R
eps_pos:eps > 0
alpg:R
lg:alpg > 0 /\
(forall x : R,
 Dg x /\ R_dist x l < alpg ->
 R_dist (g x) l' < eps)
alpg > 0
  2: tauto.
f, g:R -> R
Df, Dg:R -> Prop
l, l', x0:R
Hf:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   Df x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps0)
Hg:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   Dg x /\ R_dist x l < alp ->
   R_dist (g x) l' < eps0)
eps:R
eps_pos:eps > 0
alpg:R
lg:alpg > 0 /\
(forall x : R,
 Dg x /\ R_dist x l < alpg ->
 R_dist (g x) l' < eps)
forall x : R,
x > 0 /\
(forall x1 : R,
 Df x1 /\ R_dist x1 x0 < x -> R_dist (f x1) l < alpg) ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   (Df x1 /\ Dg (f x1)) /\ R_dist x1 x0 < alp ->
   R_dist (g (f x1)) l' < eps)
  intros alpf lf.
f, g:R -> R
Df, Dg:R -> Prop
l, l', x0:R
Hf:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   Df x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps0)
Hg:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   Dg x /\ R_dist x l < alp ->
   R_dist (g x) l' < eps0)
eps:R
eps_pos:eps > 0
alpg:R
lg:alpg > 0 /\
(forall x : R,
 Dg x /\ R_dist x l < alpg ->
 R_dist (g x) l' < eps)
alpf:R
lf:alpf > 0 /\
(forall x : R,
 Df x /\ R_dist x x0 < alpf ->
 R_dist (f x) l < alpg)
exists alp : R,
  alp > 0 /\
  (forall x : R,
   (Df x /\ Dg (f x)) /\ R_dist x x0 < alp ->
   R_dist (g (f x)) l' < eps)
  exists alpf.
f, g:R -> R
Df, Dg:R -> Prop
l, l', x0:R
Hf:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   Df x /\ R_dist x x0 < alp ->
   R_dist (f x) l < eps0)
Hg:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   Dg x /\ R_dist x l < alp ->
   R_dist (g x) l' < eps0)
eps:R
eps_pos:eps > 0
alpg:R
lg:alpg > 0 /\
(forall x : R,
 Dg x /\ R_dist x l < alpg ->
 R_dist (g x) l' < eps)
alpf:R
lf:alpf > 0 /\
(forall x : R,
 Df x /\ R_dist x x0 < alpf ->
 R_dist (f x) l < alpg)
alpf > 0 /\
(forall x : R,
 (Df x /\ Dg (f x)) /\ R_dist x x0 < alpf ->
 R_dist (g (f x)) l' < eps)
  intuition.
Qed.

(*********)

Lemma limit_inv :
  forall (f:R -> R) (D:R -> Prop) (l x0:R),
    limit1_in f D l x0 -> l <> 0 -> limit1_in (fun x:R => / f x) D (/ l) x0.
forall (f : R -> R) (D : R -> Prop) (l x0 : R),
limit1_in f D l x0 ->
l <> 0 -> limit1_in (fun x : R => / f x) D (/ l) x0
Proof.
forall (f : R -> R) (D : R -> Prop) (l x0 : R),
limit1_in f D l x0 ->
l <> 0 -> limit1_in (fun x : R => / f x) D (/ l) x0
  unfold limit1_in; unfold limit_in; simpl;
    unfold R_dist; intros; elim (H (Rabs l / 2)).
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
forall x : R,
x > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < x ->
 Rabs (f x1 - l) < Rabs l / 2) ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (/ f x1 - / l) < eps)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  intros delta1 H2; elim (H (eps * (Rsqr l / 2))).
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
forall x : R,
x > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < x ->
 Rabs (f x1 - l) < eps * (l² / 2)) ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (/ f x1 - / l) < eps)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  intros delta2 H3; elim H2; elim H3; intros; exists (Rmin delta1 delta2);
    split.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta2 ->
 Rabs (f x - l) < eps * (l² / 2))
H4:delta2 > 0
H5:forall x : R,
D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H6:delta1 > 0
H7:forall x : R,
D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
Rmin delta1 delta2 > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta2 ->
 Rabs (f x - l) < eps * (l² / 2))
H4:delta2 > 0
H5:forall x : R,
D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H6:delta1 > 0
H7:forall x : R,
D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
forall x : R,
D x /\ Rabs (x - x0) < Rmin delta1 delta2 ->
Rabs (/ f x - / l) < eps
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  unfold Rmin; case (Rle_dec delta1 delta2); intro; assumption.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta2 ->
 Rabs (f x - l) < eps * (l² / 2))
H4:delta2 > 0
H5:forall x : R,
D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H6:delta1 > 0
H7:forall x : R,
D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
forall x : R,
D x /\ Rabs (x - x0) < Rmin delta1 delta2 ->
Rabs (/ f x - / l) < eps
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  intro; generalize (H5 x); clear H5; intro H5; generalize (H7 x); clear H7;
    intro H7; intro H10; elim H10; intros; cut (D x /\ Rabs (x - x0) < delta1).
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1 ->
Rabs (/ f x - / l) < eps
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  cut (D x /\ Rabs (x - x0) < delta2).
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2 ->
D x /\ Rabs (x - x0) < delta1 ->
Rabs (/ f x - / l) < eps
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  intros; generalize (H5 H11); clear H5; intro H5; generalize (H7 H12);
    clear H7; intro H7; generalize (Rabs_triang_inv l (f x));
      intro; rewrite Rabs_minus_sym in H7;
        generalize
          (Rle_lt_trans (Rabs l - Rabs (f x)) (Rabs (l - f x)) (Rabs l / 2) H13 H7);
          intro;
            generalize
              (Rplus_lt_compat_l (Rabs (f x) - Rabs l / 2) (Rabs l - Rabs (f x))
                (Rabs l / 2) H14);
              replace (Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))) with (Rabs l / 2).
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 < Rabs (f x) - Rabs l / 2 + Rabs l / 2 ->
Rabs (/ f x - / l) < eps
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  unfold Rminus; rewrite Rplus_assoc; rewrite Rplus_opp_l;
    rewrite Rplus_0_r; intro; cut (f x <> 0).
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0 -> Rabs (/ f x + - / l) < eps
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  intro; replace (/ f x + - / l) with ((l - f x) * / (l * f x)).
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs ((l - f x) * / (l * f x)) < eps
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  rewrite Rabs_mult; rewrite Rabs_Rinv.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (l - f x) * / Rabs (l * f x) < eps
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  cut (/ Rabs (l * f x) < 2 / Rsqr l).
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
/ Rabs (l * f x) < 2 / l² ->
Rabs (l - f x) * / Rabs (l * f x) < eps
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
/ Rabs (l * f x) < 2 / l²
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  intro; rewrite Rabs_minus_sym in H5; cut (0 <= / Rabs (l * f x)).
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
0 <= / Rabs (l * f x) ->
Rabs (l - f x) * / Rabs (l * f x) < eps
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
0 <= / Rabs (l * f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
/ Rabs (l * f x) < 2 / l²
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  intro;
    generalize
      (Rmult_le_0_lt_compat (Rabs (l - f x)) (eps * (Rsqr l / 2))
        (/ Rabs (l * f x)) (2 / Rsqr l) (Rabs_pos (l - f x)) H18 H5 H17);
      replace (eps * (Rsqr l / 2) * (2 / Rsqr l)) with eps.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
Rabs (l - f x) * / Rabs (l * f x) < eps ->
Rabs (l - f x) * / Rabs (l * f x) < eps
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
eps = eps * (l² / 2) * (2 / l²)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
0 <= / Rabs (l * f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
/ Rabs (l * f x) < 2 / l²
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  intro; assumption.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
eps = eps * (l² / 2) * (2 / l²)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
0 <= / Rabs (l * f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
/ Rabs (l * f x) < 2 / l²
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  unfold Rdiv; unfold Rsqr; rewrite Rinv_mult_distr.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
eps = eps * (l * l * / 2) * (2 * (/ l * / l))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
0 <= / Rabs (l * f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
/ Rabs (l * f x) < 2 / l²
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  repeat rewrite Rmult_assoc.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
eps = eps * (l * (l * (/ 2 * (2 * (/ l * / l)))))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
0 <= / Rabs (l * f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
/ Rabs (l * f x) < 2 / l²
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  rewrite (Rmult_comm l).
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
eps = eps * (l * (/ 2 * (2 * (/ l * / l))) * l)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
0 <= / Rabs (l * f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
/ Rabs (l * f x) < 2 / l²
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  repeat rewrite Rmult_assoc.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
eps = eps * (l * (/ 2 * (2 * (/ l * (/ l * l)))))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
0 <= / Rabs (l * f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
/ Rabs (l * f x) < 2 / l²
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  rewrite <- Rinv_l_sym.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
eps = eps * (l * (/ 2 * (2 * (/ l * 1))))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
0 <= / Rabs (l * f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
/ Rabs (l * f x) < 2 / l²
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  rewrite Rmult_1_r.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
eps = eps * (l * (/ 2 * (2 * / l)))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
0 <= / Rabs (l * f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
/ Rabs (l * f x) < 2 / l²
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  rewrite (Rmult_comm l).
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
eps = eps * (/ 2 * (2 * / l) * l)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
0 <= / Rabs (l * f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
/ Rabs (l * f x) < 2 / l²
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  repeat rewrite Rmult_assoc.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
eps = eps * (/ 2 * (2 * (/ l * l)))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
0 <= / Rabs (l * f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
/ Rabs (l * f x) < 2 / l²
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  rewrite <- Rinv_l_sym.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
eps = eps * (/ 2 * (2 * 1))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
0 <= / Rabs (l * f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
/ Rabs (l * f x) < 2 / l²
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  rewrite Rmult_1_r.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
eps = eps * (/ 2 * 2)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
0 <= / Rabs (l * f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
/ Rabs (l * f x) < 2 / l²
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  rewrite <- Rinv_l_sym.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
eps = eps * 1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
2 <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
0 <= / Rabs (l * f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
/ Rabs (l * f x) < 2 / l²
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  rewrite Rmult_1_r; reflexivity.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
2 <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
0 <= / Rabs (l * f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
/ Rabs (l * f x) < 2 / l²
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  discrR.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
0 <= / Rabs (l * f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
/ Rabs (l * f x) < 2 / l²
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  exact H0.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
0 <= / Rabs (l * f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
/ Rabs (l * f x) < 2 / l²
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  exact H0.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
0 <= / Rabs (l * f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
/ Rabs (l * f x) < 2 / l²
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  exact H0.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
H18:0 <= / Rabs (l * f x)
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
0 <= / Rabs (l * f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
/ Rabs (l * f x) < 2 / l²
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  exact H0.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (l - f x) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
H17:/ Rabs (l * f x) < 2 / l²
0 <= / Rabs (l * f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
/ Rabs (l * f x) < 2 / l²
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  left; apply Rinv_0_lt_compat; apply Rabs_pos_lt; apply prod_neq_R0;
    assumption.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
/ Rabs (l * f x) < 2 / l²
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  rewrite Rmult_comm; rewrite Rabs_mult; rewrite Rinv_mult_distr.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
/ Rabs (f x) * / Rabs l < 2 / l²
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  rewrite (Rsqr_abs l); unfold Rsqr; unfold Rdiv;
    rewrite Rinv_mult_distr.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
/ Rabs (f x) * / Rabs l < 2 * (/ Rabs l * / Rabs l)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  repeat rewrite <- Rmult_assoc; apply Rmult_lt_compat_r.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
0 < / Rabs l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
/ Rabs (f x) < 2 * / Rabs l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
/ Rabs (f x) < 2 * / Rabs l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  apply Rmult_lt_reg_l with (Rabs (f x) * Rabs l * / 2).
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
0 < Rabs (f x) * Rabs l * / 2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) * Rabs l * / 2 * / Rabs (f x) <
Rabs (f x) * Rabs l * / 2 * (2 * / Rabs l)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  repeat apply Rmult_lt_0_compat.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
0 < Rabs (f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
0 < Rabs l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
0 < / 2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) * Rabs l * / 2 * / Rabs (f x) <
Rabs (f x) * Rabs l * / 2 * (2 * / Rabs l)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  apply Rabs_pos_lt; assumption.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
0 < Rabs l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
0 < / 2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) * Rabs l * / 2 * / Rabs (f x) <
Rabs (f x) * Rabs l * / 2 * (2 * / Rabs l)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  apply Rabs_pos_lt; assumption.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
0 < / 2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) * Rabs l * / 2 * / Rabs (f x) <
Rabs (f x) * Rabs l * / 2 * (2 * / Rabs l)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  apply Rinv_0_lt_compat; cut (0%nat <> 2%nat);
    [ intro H17; generalize (lt_INR_0 2 (neq_O_lt 2 H17)); unfold INR;
      intro H18; assumption
      | discriminate ].
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) * Rabs l * / 2 * / Rabs (f x) <
Rabs (f x) * Rabs l * / 2 * (2 * / Rabs l)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  replace (Rabs (f x) * Rabs l * / 2 * / Rabs (f x)) with (Rabs l / 2).
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l / 2 <
Rabs (f x) * Rabs l * / 2 * (2 * / Rabs l)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l / 2 = Rabs (f x) * Rabs l * / 2 * / Rabs (f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  replace (Rabs (f x) * Rabs l * / 2 * (2 * / Rabs l)) with (Rabs (f x)).
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l / 2 < Rabs (f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) =
Rabs (f x) * Rabs l * / 2 * (2 * / Rabs l)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l / 2 = Rabs (f x) * Rabs l * / 2 * / Rabs (f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  assumption.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) =
Rabs (f x) * Rabs l * / 2 * (2 * / Rabs l)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l / 2 = Rabs (f x) * Rabs l * / 2 * / Rabs (f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  repeat rewrite Rmult_assoc.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) =
Rabs (f x) * (Rabs l * (/ 2 * (2 * / Rabs l)))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l / 2 = Rabs (f x) * Rabs l * / 2 * / Rabs (f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  rewrite (Rmult_comm (Rabs l)).
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) =
Rabs (f x) * (/ 2 * (2 * / Rabs l) * Rabs l)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l / 2 = Rabs (f x) * Rabs l * / 2 * / Rabs (f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  repeat rewrite Rmult_assoc.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) =
Rabs (f x) * (/ 2 * (2 * (/ Rabs l * Rabs l)))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l / 2 = Rabs (f x) * Rabs l * / 2 * / Rabs (f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  rewrite <- Rinv_l_sym.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) = Rabs (f x) * (/ 2 * (2 * 1))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l / 2 = Rabs (f x) * Rabs l * / 2 * / Rabs (f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  rewrite Rmult_1_r.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) = Rabs (f x) * (/ 2 * 2)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l / 2 = Rabs (f x) * Rabs l * / 2 * / Rabs (f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  rewrite <- Rinv_l_sym.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) = Rabs (f x) * 1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
2 <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l / 2 = Rabs (f x) * Rabs l * / 2 * / Rabs (f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  rewrite Rmult_1_r; reflexivity.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
2 <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l / 2 = Rabs (f x) * Rabs l * / 2 * / Rabs (f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  discrR.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l / 2 = Rabs (f x) * Rabs l * / 2 * / Rabs (f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  apply Rabs_no_R0.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l / 2 = Rabs (f x) * Rabs l * / 2 * / Rabs (f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  assumption.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l / 2 = Rabs (f x) * Rabs l * / 2 * / Rabs (f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  unfold Rdiv.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l * / 2 =
Rabs (f x) * Rabs l * / 2 * / Rabs (f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  repeat rewrite Rmult_assoc.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l * / 2 =
Rabs (f x) * (Rabs l * (/ 2 * / Rabs (f x)))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  rewrite (Rmult_comm (Rabs (f x))).
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l * / 2 =
Rabs l * (/ 2 * / Rabs (f x)) * Rabs (f x)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  repeat rewrite Rmult_assoc.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l * / 2 =
Rabs l * (/ 2 * (/ Rabs (f x) * Rabs (f x)))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  rewrite <- Rinv_l_sym.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l * / 2 = Rabs l * (/ 2 * 1)
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  rewrite Rmult_1_r.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l * / 2 = Rabs l * / 2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  reflexivity.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  apply Rabs_no_R0; assumption.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  apply Rabs_no_R0; assumption.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  apply Rabs_no_R0; assumption.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs (f x) <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  apply Rabs_no_R0; assumption.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
Rabs l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  apply Rabs_no_R0; assumption.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  apply prod_neq_R0; assumption.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * / (l * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  rewrite (Rinv_mult_distr _ _ H0 H16).
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
(l - f x) * (/ l * / f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  unfold Rminus; rewrite Rmult_plus_distr_r.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * (/ l * / f x) + - f x * (/ l * / f x) =
/ f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  rewrite <- Rmult_assoc.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l * / l * / f x + - f x * (/ l * / f x) =
/ f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  rewrite <- Rinv_r_sym.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
1 * / f x + - f x * (/ l * / f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  rewrite Rmult_1_l.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
/ f x + - f x * (/ l * / f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  rewrite Ropp_mult_distr_l_reverse.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
/ f x + - (f x * (/ l * / f x)) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  rewrite (Rmult_comm (f x)).
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
/ f x + - (/ l * / f x * f x) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  rewrite Rmult_assoc.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
/ f x + - (/ l * (/ f x * f x)) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  rewrite <- Rinv_l_sym.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
/ f x + - (/ l * 1) = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  rewrite Rmult_1_r.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
/ f x + - / l = / f x + - / l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  reflexivity.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  assumption.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
H16:f x <> 0
l <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  assumption.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < Rabs (f x)
f x <> 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  red; intro; rewrite H16 in H15; rewrite Rabs_R0 in H15;
    cut (0 < Rabs l / 2).
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < 0
H16:f x = 0
0 < Rabs l / 2 -> False
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < 0
H16:f x = 0
0 < Rabs l / 2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  intro; elim (Rlt_irrefl 0 (Rlt_trans 0 (Rabs l / 2) 0 H17 H15)).
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < 0
H16:f x = 0
0 < Rabs l / 2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  unfold Rdiv; apply Rmult_lt_0_compat.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < 0
H16:f x = 0
0 < Rabs l
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < 0
H16:f x = 0
0 < / 2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  apply Rabs_pos_lt; assumption.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
H15:Rabs l / 2 < 0
H16:f x = 0
0 < / 2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  apply Rinv_0_lt_compat; cut (0%nat <> 2%nat);
    [ intro H17; generalize (lt_INR_0 2 (neq_O_lt 2 H17)); unfold INR;
      intro; assumption
      | discriminate ].
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  pattern (Rabs l) at 3; rewrite double_var.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
H11:D x /\ Rabs (x - x0) < delta2
H12:D x /\ Rabs (x - x0) < delta1
H5:Rabs (f x - l) < eps * (l² / 2)
H7:Rabs (l - f x) < Rabs l / 2
H13:Rabs l - Rabs (f x) <= Rabs (l - f x)
H14:Rabs l - Rabs (f x) < Rabs l / 2
Rabs l / 2 =
Rabs (f x) - Rabs l / 2 +
(Rabs l / 2 + Rabs l / 2 - Rabs (f x))
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  ring.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  split;
    [ assumption
      | apply Rlt_le_trans with (Rmin delta1 delta2);
        [ assumption | apply Rmin_r ] ].
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D x1 /\ Rabs (x1 - x0) < alp ->
   Rabs (f x1 - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta1 ->
 Rabs (f x1 - l) < Rabs l / 2)
delta2:R
H3:delta2 > 0 /\
(forall x1 : R,
 D x1 /\ Rabs (x1 - x0) < delta2 ->
 Rabs (f x1 - l) < eps * (l² / 2))
H4:delta2 > 0
H6:delta1 > 0
x:R
H5:D x /\ Rabs (x - x0) < delta2 ->
Rabs (f x - l) < eps * (l² / 2)
H7:D x /\ Rabs (x - x0) < delta1 ->
Rabs (f x - l) < Rabs l / 2
H10:D x /\ Rabs (x - x0) < Rmin delta1 delta2
H8:D x
H9:Rabs (x - x0) < Rmin delta1 delta2
D x /\ Rabs (x - x0) < delta1
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  split;
    [ assumption
      | apply Rlt_le_trans with (Rmin delta1 delta2);
        [ assumption | apply Rmin_l ] ].
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
eps * (l² / 2) > 0
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  change (0 < eps * (Rsqr l / 2)); unfold Rdiv;
    repeat rewrite Rmult_assoc; apply Rmult_lt_0_compat.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
0 < eps
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
0 < l² * / 2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  assumption.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
0 < l² * / 2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  apply Rmult_lt_0_compat.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
0 < l²
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
0 < / 2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0 apply Rsqr_pos_lt; assumption.
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
delta1:R
H2:delta1 > 0 /\
(forall x : R,
 D x /\ Rabs (x - x0) < delta1 ->
 Rabs (f x - l) < Rabs l / 2)
0 < / 2
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  apply Rinv_0_lt_compat; cut (0%nat <> 2%nat);
    [ intro H3; generalize (lt_INR_0 2 (neq_O_lt 2 H3)); unfold INR;
      intro; assumption
      | discriminate ].
f:R -> R
D:R -> Prop
l, x0:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D x /\ Rabs (x - x0) < alp ->
   Rabs (f x - l) < eps0)
H0:l <> 0
eps:R
H1:eps > 0
Rabs l / 2 > 0
  change (0 < Rabs l / 2); unfold Rdiv; apply Rmult_lt_0_compat;
    [ apply Rabs_pos_lt; assumption
      | apply Rinv_0_lt_compat; cut (0%nat <> 2%nat);
        [ intro H3; generalize (lt_INR_0 2 (neq_O_lt 2 H3)); unfold INR;
          intro; assumption
          | discriminate ] ].
Qed.