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

Require Import Rbase.
Require Import Rfunctions.
Require Export Rlimit.
Require Export Rderiv.
Local Open Scope R_scope.
Implicit Type f : R -> R.

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

Basic operations on functions

(****************************************************)
Definition plus_fct f1 f2 (x:R) : R := f1 x + f2 x.
Definition opp_fct f (x:R) : R := - f x.
Definition mult_fct f1 f2 (x:R) : R := f1 x * f2 x.
Definition mult_real_fct (a:R) f (x:R) : R := a * f x.
Definition minus_fct f1 f2 (x:R) : R := f1 x - f2 x.
Definition div_fct f1 f2 (x:R) : R := f1 x / f2 x.
Definition div_real_fct (a:R) f (x:R) : R := a / f x.
Definition comp f1 f2 (x:R) : R := f1 (f2 x).
Definition inv_fct f (x:R) : R := / f x.

Declare Scope Rfun_scope.
Delimit Scope Rfun_scope with F.

Arguments plus_fct (f1 f2)%F x%R.
Arguments mult_fct (f1 f2)%F x%R.
Arguments minus_fct (f1 f2)%F x%R.
Arguments div_fct (f1 f2)%F x%R.
Arguments inv_fct f%F x%R.
Arguments opp_fct f%F x%R.
Arguments mult_real_fct a%R f%F x%R.
Arguments div_real_fct a%R f%F x%R.
Arguments comp (f1 f2)%F x%R.

Infix "+" := plus_fct : Rfun_scope.
Notation "- x" := (opp_fct x) : Rfun_scope.
Infix "*" := mult_fct : Rfun_scope.
Infix "-" := minus_fct : Rfun_scope.
Infix "/" := div_fct : Rfun_scope.
Local Notation "f1 'o' f2" := (comp f1 f2)
  (at level 20, right associativity) : Rfun_scope.
Notation "/ x" := (inv_fct x) : Rfun_scope.

Definition fct_cte (a x:R) : R := a.
Definition id (x:R) := x.

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

Variations of functions

(****************************************************)
Definition increasing f : Prop := forall x y:R, x <= y -> f x <= f y.
Definition decreasing f : Prop := forall x y:R, x <= y -> f y <= f x.
Definition strict_increasing f : Prop := forall x y:R, x < y -> f x < f y.
Definition strict_decreasing f : Prop := forall x y:R, x < y -> f y < f x.
Definition constant f : Prop := forall x y:R, f x = f y.

(**********)
Definition no_cond (x:R) : Prop := True.

(**********)
Definition constant_D_eq f (D:R -> Prop) (c:R) : Prop :=
  forall x:R, D x -> f x = c.

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

Definition of continuity as a limit

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

(**********)
Definition continuity_pt f (x0:R) : Prop := continue_in f no_cond x0.
Definition continuity f : Prop := forall x:R, continuity_pt f x.

Arguments continuity_pt f%F x0%R.
Arguments continuity f%F.

Lemma continuity_pt_locally_ext :
  forall f g a x, 0 < a -> (forall y, R_dist y x < a -> f y = g y) ->
  continuity_pt f x -> continuity_pt g x.forall (f g : R -> R) (a x : R),
0 < a ->
(forall y : R, R_dist y x < a -> f y = g y) ->
continuity_pt f x -> continuity_pt g x
intros f g a x a0 q cf eps ep.f, g:R -> R
a, x:R
a0:0 < a
q:forall y : R, R_dist y x < a -> f y = g y
cf:continuity_pt f x
eps:R
ep:eps > 0
exists alp : R,
  alp > 0 /\
  (forall x0 : Base R_met,
   D_x no_cond x x0 /\ dist R_met x0 x < alp ->
   dist R_met (g x0) (g x) < eps)
destruct (cf eps ep) as [a' [a'p Pa']].f, g:R -> R
a, x:R
a0:0 < a
q:forall y : R, R_dist y x < a -> f y = g y
cf:continuity_pt f x
eps:R
ep:eps > 0
a':R
a'p:a' > 0
Pa':forall x0 : Base R_met,
D_x no_cond x x0 /\ dist R_met x0 x < a' -> dist R_met (f x0) (f x) < eps
exists alp : R,
  alp > 0 /\
  (forall x0 : Base R_met,
   D_x no_cond x x0 /\ dist R_met x0 x < alp ->
   dist R_met (g x0) (g x) < eps)
exists (Rmin a a'); split.f, g:R -> R
a, x:R
a0:0 < a
q:forall y : R, R_dist y x < a -> f y = g y
cf:continuity_pt f x
eps:R
ep:eps > 0
a':R
a'p:a' > 0
Pa':forall x0 : Base R_met,
D_x no_cond x x0 /\ dist R_met x0 x < a' -> dist R_met (f x0) (f x) < eps
Rmin a a' > 0
f, g:R -> R
a, x:R
a0:0 < a
q:forall y : R, R_dist y x < a -> f y = g y
cf:continuity_pt f x
eps:R
ep:eps > 0
a':R
a'p:a' > 0
Pa':forall x0 : Base R_met,
D_x no_cond x x0 /\ dist R_met x0 x < a' -> dist R_met (f x0) (f x) < eps
forall x0 : Base R_met,
D_x no_cond x x0 /\ dist R_met x0 x < Rmin a a' ->
dist R_met (g x0) (g x) < eps
 unfold Rmin; destruct (Rle_dec a a').f, g:R -> R
a, x:R
a0:0 < a
q:forall y : R, R_dist y x < a -> f y = g y
cf:continuity_pt f x
eps:R
ep:eps > 0
a':R
a'p:a' > 0
Pa':forall x0 : Base R_met,
D_x no_cond x x0 /\ dist R_met x0 x < a' -> dist R_met (f x0) (f x) < eps
r:a <= a'
a > 0
f, g:R -> R
a, x:R
a0:0 < a
q:forall y : R, R_dist y x < a -> f y = g y
cf:continuity_pt f x
eps:R
ep:eps > 0
a':R
a'p:a' > 0
Pa':forall x0 : Base R_met,
D_x no_cond x x0 /\ dist R_met x0 x < a' -> dist R_met (f x0) (f x) < eps
n:~ a <= a'
a' > 0
f, g:R -> R
a, x:R
a0:0 < a
q:forall y : R, R_dist y x < a -> f y = g y
cf:continuity_pt f x
eps:R
ep:eps > 0
a':R
a'p:a' > 0
Pa':forall x0 : Base R_met,
D_x no_cond x x0 /\ dist R_met x0 x < a' -> dist R_met (f x0) (f x) < eps
forall x0 : Base R_met,
D_x no_cond x x0 /\ dist R_met x0 x < Rmin a a' ->
dist R_met (g x0) (g x) < eps
  assumption.f, g:R -> R
a, x:R
a0:0 < a
q:forall y : R, R_dist y x < a -> f y = g y
cf:continuity_pt f x
eps:R
ep:eps > 0
a':R
a'p:a' > 0
Pa':forall x0 : Base R_met,
D_x no_cond x x0 /\ dist R_met x0 x < a' -> dist R_met (f x0) (f x) < eps
n:~ a <= a'
a' > 0
f, g:R -> R
a, x:R
a0:0 < a
q:forall y : R, R_dist y x < a -> f y = g y
cf:continuity_pt f x
eps:R
ep:eps > 0
a':R
a'p:a' > 0
Pa':forall x0 : Base R_met,
D_x no_cond x x0 /\ dist R_met x0 x < a' -> dist R_met (f x0) (f x) < eps
forall x0 : Base R_met,
D_x no_cond x x0 /\ dist R_met x0 x < Rmin a a' ->
dist R_met (g x0) (g x) < eps
 assumption.f, g:R -> R
a, x:R
a0:0 < a
q:forall y : R, R_dist y x < a -> f y = g y
cf:continuity_pt f x
eps:R
ep:eps > 0
a':R
a'p:a' > 0
Pa':forall x0 : Base R_met,
D_x no_cond x x0 /\ dist R_met x0 x < a' -> dist R_met (f x0) (f x) < eps
forall x0 : Base R_met,
D_x no_cond x x0 /\ dist R_met x0 x < Rmin a a' ->
dist R_met (g x0) (g x) < eps
intros y cy; rewrite <- !q.f, g:R -> R
a, x:R
a0:0 < a
q:forall y0 : R, R_dist y0 x < a -> f y0 = g y0
cf:continuity_pt f x
eps:R
ep:eps > 0
a':R
a'p:a' > 0
Pa':forall x0 : Base R_met,
D_x no_cond x x0 /\ dist R_met x0 x < a' -> dist R_met (f x0) (f x) < eps
y:Base R_met
cy:D_x no_cond x y /\ dist R_met y x < Rmin a a'
dist R_met (f y) (f x) < eps
f, g:R -> R
a, x:R
a0:0 < a
q:forall y0 : R, R_dist y0 x < a -> f y0 = g y0
cf:continuity_pt f x
eps:R
ep:eps > 0
a':R
a'p:a' > 0
Pa':forall x0 : Base R_met,
D_x no_cond x x0 /\ dist R_met x0 x < a' -> dist R_met (f x0) (f x) < eps
y:Base R_met
cy:D_x no_cond x y /\ dist R_met y x < Rmin a a'
R_dist x x < a
f, g:R -> R
a, x:R
a0:0 < a
q:forall y0 : R, R_dist y0 x < a -> f y0 = g y0
cf:continuity_pt f x
eps:R
ep:eps > 0
a':R
a'p:a' > 0
Pa':forall x0 : Base R_met,
D_x no_cond x x0 /\ dist R_met x0 x < a' -> dist R_met (f x0) (f x) < eps
y:Base R_met
cy:D_x no_cond x y /\ dist R_met y x < Rmin a a'
R_dist y x < a
  apply Pa'.f, g:R -> R
a, x:R
a0:0 < a
q:forall y0 : R, R_dist y0 x < a -> f y0 = g y0
cf:continuity_pt f x
eps:R
ep:eps > 0
a':R
a'p:a' > 0
Pa':forall x0 : Base R_met,
D_x no_cond x x0 /\ dist R_met x0 x < a' -> dist R_met (f x0) (f x) < eps
y:Base R_met
cy:D_x no_cond x y /\ dist R_met y x < Rmin a a'
D_x no_cond x y /\ dist R_met y x < a'
f, g:R -> R
a, x:R
a0:0 < a
q:forall y0 : R, R_dist y0 x < a -> f y0 = g y0
cf:continuity_pt f x
eps:R
ep:eps > 0
a':R
a'p:a' > 0
Pa':forall x0 : Base R_met,
D_x no_cond x x0 /\ dist R_met x0 x < a' -> dist R_met (f x0) (f x) < eps
y:Base R_met
cy:D_x no_cond x y /\ dist R_met y x < Rmin a a'
R_dist x x < a
f, g:R -> R
a, x:R
a0:0 < a
q:forall y0 : R, R_dist y0 x < a -> f y0 = g y0
cf:continuity_pt f x
eps:R
ep:eps > 0
a':R
a'p:a' > 0
Pa':forall x0 : Base R_met,
D_x no_cond x x0 /\ dist R_met x0 x < a' -> dist R_met (f x0) (f x) < eps
y:Base R_met
cy:D_x no_cond x y /\ dist R_met y x < Rmin a a'
R_dist y x < a
  split;[| apply Rlt_le_trans with (Rmin a a');[ | apply Rmin_r]];tauto.f, g:R -> R
a, x:R
a0:0 < a
q:forall y0 : R, R_dist y0 x < a -> f y0 = g y0
cf:continuity_pt f x
eps:R
ep:eps > 0
a':R
a'p:a' > 0
Pa':forall x0 : Base R_met,
D_x no_cond x x0 /\ dist R_met x0 x < a' -> dist R_met (f x0) (f x) < eps
y:Base R_met
cy:D_x no_cond x y /\ dist R_met y x < Rmin a a'
R_dist x x < a
f, g:R -> R
a, x:R
a0:0 < a
q:forall y0 : R, R_dist y0 x < a -> f y0 = g y0
cf:continuity_pt f x
eps:R
ep:eps > 0
a':R
a'p:a' > 0
Pa':forall x0 : Base R_met,
D_x no_cond x x0 /\ dist R_met x0 x < a' -> dist R_met (f x0) (f x) < eps
y:Base R_met
cy:D_x no_cond x y /\ dist R_met y x < Rmin a a'
R_dist y x < a
 rewrite R_dist_eq; assumption.f, g:R -> R
a, x:R
a0:0 < a
q:forall y0 : R, R_dist y0 x < a -> f y0 = g y0
cf:continuity_pt f x
eps:R
ep:eps > 0
a':R
a'p:a' > 0
Pa':forall x0 : Base R_met,
D_x no_cond x x0 /\ dist R_met x0 x < a' -> dist R_met (f x0) (f x) < eps
y:Base R_met
cy:D_x no_cond x y /\ dist R_met y x < Rmin a a'
R_dist y x < a   
apply Rlt_le_trans with (Rmin a a');[ | apply Rmin_l]; tauto.
Qed.


(**********)
Lemma continuity_pt_plus :
  forall f1 f2 (x0:R),
    continuity_pt f1 x0 -> continuity_pt f2 x0 -> continuity_pt (f1 + f2) x0.forall (f1 f2 : R -> R) (x0 : R),
continuity_pt f1 x0 ->
continuity_pt f2 x0 -> continuity_pt (f1 + f2) x0
Proof.forall (f1 f2 : R -> R) (x0 : R),
continuity_pt f1 x0 ->
continuity_pt f2 x0 -> continuity_pt (f1 + f2) x0
  unfold continuity_pt, plus_fct; unfold continue_in; intros;
    apply limit_plus; assumption.
Qed.

Lemma continuity_pt_opp :
  forall f (x0:R), continuity_pt f x0 -> continuity_pt (- f) x0.forall (f : R -> R) (x0 : R),
continuity_pt f x0 -> continuity_pt (- f) x0
Proof.forall (f : R -> R) (x0 : R),
continuity_pt f x0 -> continuity_pt (- f) x0
  unfold continuity_pt, opp_fct; unfold continue_in; intros;
    apply limit_Ropp; assumption.
Qed.

Lemma continuity_pt_minus :
  forall f1 f2 (x0:R),
    continuity_pt f1 x0 -> continuity_pt f2 x0 -> continuity_pt (f1 - f2) x0.forall (f1 f2 : R -> R) (x0 : R),
continuity_pt f1 x0 ->
continuity_pt f2 x0 -> continuity_pt (f1 - f2) x0
Proof.forall (f1 f2 : R -> R) (x0 : R),
continuity_pt f1 x0 ->
continuity_pt f2 x0 -> continuity_pt (f1 - f2) x0
  unfold continuity_pt, minus_fct; unfold continue_in; intros;
    apply limit_minus; assumption.
Qed.

Lemma continuity_pt_mult :
  forall f1 f2 (x0:R),
    continuity_pt f1 x0 -> continuity_pt f2 x0 -> continuity_pt (f1 * f2) x0.forall (f1 f2 : R -> R) (x0 : R),
continuity_pt f1 x0 ->
continuity_pt f2 x0 -> continuity_pt (f1 * f2) x0
Proof.forall (f1 f2 : R -> R) (x0 : R),
continuity_pt f1 x0 ->
continuity_pt f2 x0 -> continuity_pt (f1 * f2) x0
  unfold continuity_pt, mult_fct; unfold continue_in; intros;
    apply limit_mul; assumption.
Qed.

Lemma continuity_pt_const : forall f (x0:R), constant f -> continuity_pt f x0.forall (f : R -> R) (x0 : R),
constant f -> continuity_pt f x0
Proof.forall (f : R -> R) (x0 : R),
constant f -> continuity_pt f x0
  unfold constant, continuity_pt; unfold continue_in;
    unfold limit1_in; unfold limit_in;
      intros; exists 1; split;
        [ apply Rlt_0_1
          | intros; generalize (H x x0); intro; rewrite H2; simpl;
            rewrite R_dist_eq; assumption ].
Qed.

Lemma continuity_pt_scal :
  forall f (a x0:R),
    continuity_pt f x0 -> continuity_pt (mult_real_fct a f) x0.forall (f : R -> R) (a x0 : R),
continuity_pt f x0 ->
continuity_pt (mult_real_fct a f) x0
Proof.forall (f : R -> R) (a x0 : R),
continuity_pt f x0 ->
continuity_pt (mult_real_fct a f) x0
  unfold continuity_pt, mult_real_fct; unfold continue_in;
    intros; apply (limit_mul (fun x:R => a) f (D_x no_cond x0) a (f x0) x0).f:R -> R
a, x0:R
H:limit1_in f (D_x no_cond x0) (f x0) x0
limit1_in (fun _ : R => a) (D_x no_cond x0) a x0
f:R -> R
a, x0:R
H:limit1_in f (D_x no_cond x0) (f x0) x0
limit1_in f (D_x no_cond x0) (f x0) x0
  unfold limit1_in; unfold limit_in; intros; exists 1; split.f:R -> R
a, x0:R
H:limit1_in f (D_x no_cond x0) (f x0) x0
eps:R
H0:eps > 0
1 > 0
f:R -> R
a, x0:R
H:limit1_in f (D_x no_cond x0) (f x0) x0
eps:R
H0:eps > 0
forall x : Base R_met,
D_x no_cond x0 x /\ dist R_met x x0 < 1 ->
dist R_met a a < eps
f:R -> R
a, x0:R
H:limit1_in f (D_x no_cond x0) (f x0) x0
limit1_in f (D_x no_cond x0) (f x0) x0
  apply Rlt_0_1.f:R -> R
a, x0:R
H:limit1_in f (D_x no_cond x0) (f x0) x0
eps:R
H0:eps > 0
forall x : Base R_met,
D_x no_cond x0 x /\ dist R_met x x0 < 1 ->
dist R_met a a < eps
f:R -> R
a, x0:R
H:limit1_in f (D_x no_cond x0) (f x0) x0
limit1_in f (D_x no_cond x0) (f x0) x0
  intros; rewrite R_dist_eq; assumption.f:R -> R
a, x0:R
H:limit1_in f (D_x no_cond x0) (f x0) x0
limit1_in f (D_x no_cond x0) (f x0) x0
  assumption.
Qed.

Lemma continuity_pt_inv :
  forall f (x0:R), continuity_pt f x0 -> f x0 <> 0 -> continuity_pt (/ f) x0.forall (f : R -> R) (x0 : R),
continuity_pt f x0 ->
f x0 <> 0 -> continuity_pt (/ f) x0
Proof.forall (f : R -> R) (x0 : R),
continuity_pt f x0 ->
f x0 <> 0 -> continuity_pt (/ f) x0
  intros.f:R -> R
x0:R
H:continuity_pt f x0
H0:f x0 <> 0
continuity_pt (/ f) x0
  replace (/ f)%F with (fun x:R => / f x).f:R -> R
x0:R
H:continuity_pt f x0
H0:f x0 <> 0
continuity_pt (fun x : R => / f x) x0
f:R -> R
x0:R
H:continuity_pt f x0
H0:f x0 <> 0
(fun x : R => / f x) = (/ f)%F
  unfold continuity_pt; unfold continue_in; intros;
    apply limit_inv; assumption.f:R -> R
x0:R
H:continuity_pt f x0
H0:f x0 <> 0
(fun x : R => / f x) = (/ f)%F
  unfold inv_fct; reflexivity.
Qed.

Lemma div_eq_inv : forall f1 f2, (f1 / f2)%F = (f1 * / f2)%F.forall f1 f2 : R -> R, (f1 / f2)%F = (f1 * / f2)%F
Proof.forall f1 f2 : R -> R, (f1 / f2)%F = (f1 * / f2)%F
  intros; reflexivity.
Qed.

Lemma continuity_pt_div :
  forall f1 f2 (x0:R),
    continuity_pt f1 x0 ->
    continuity_pt f2 x0 -> f2 x0 <> 0 -> continuity_pt (f1 / f2) x0.forall (f1 f2 : R -> R) (x0 : R),
continuity_pt f1 x0 ->
continuity_pt f2 x0 ->
f2 x0 <> 0 -> continuity_pt (f1 / f2) x0
Proof.forall (f1 f2 : R -> R) (x0 : R),
continuity_pt f1 x0 ->
continuity_pt f2 x0 ->
f2 x0 <> 0 -> continuity_pt (f1 / f2) x0
  intros; rewrite (div_eq_inv f1 f2); apply continuity_pt_mult;
    [ assumption | apply continuity_pt_inv; assumption ].
Qed.

Lemma continuity_pt_comp :
  forall f1 f2 (x:R),
    continuity_pt f1 x -> continuity_pt f2 (f1 x) -> continuity_pt (f2 o f1) x.forall (f1 f2 : R -> R) (x : R),
continuity_pt f1 x ->
continuity_pt f2 (f1 x) -> continuity_pt (f2 o f1) x
Proof.forall (f1 f2 : R -> R) (x : R),
continuity_pt f1 x ->
continuity_pt f2 (f1 x) -> continuity_pt (f2 o f1) x
  unfold continuity_pt; unfold continue_in; intros;
    unfold comp.f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
limit1_in (fun x0 : R => f2 (f1 x0)) (D_x no_cond x)
  (f2 (f1 x)) x
  cut
    (limit1_in (fun x0:R => f2 (f1 x0))
      (Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1) (
        f2 (f1 x)) x ->
      limit1_in (fun x0:R => f2 (f1 x0)) (D_x no_cond x) (f2 (f1 x)) x).f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
(limit1_in (fun x0 : R => f2 (f1 x0))
   (Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1)
   (f2 (f1 x)) x ->
 limit1_in (fun x0 : R => f2 (f1 x0)) (D_x no_cond x)
   (f2 (f1 x)) x) ->
limit1_in (fun x0 : R => f2 (f1 x0)) (D_x no_cond x)
  (f2 (f1 x)) x
f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
limit1_in (fun x0 : R => f2 (f1 x0))
  (Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1)
  (f2 (f1 x)) x ->
limit1_in (fun x0 : R => f2 (f1 x0)) (D_x no_cond x)
  (f2 (f1 x)) x
  intro; apply H1.f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:limit1_in (fun x0 : R => f2 (f1 x0))
  (Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1)
  (f2 (f1 x)) x ->
limit1_in (fun x0 : R => f2 (f1 x0))
  (D_x no_cond x) (f2 (f1 x)) x
limit1_in (fun x0 : R => f2 (f1 x0))
  (Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1)
  (f2 (f1 x)) x
f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
limit1_in (fun x0 : R => f2 (f1 x0))
  (Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1)
  (f2 (f1 x)) x ->
limit1_in (fun x0 : R => f2 (f1 x0)) (D_x no_cond x)
  (f2 (f1 x)) x
  eapply limit_comp.f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:limit1_in (fun x0 : R => f2 (f1 x0))
  (Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1)
  (f2 (f1 x)) x ->
limit1_in (fun x0 : R => f2 (f1 x0))
  (D_x no_cond x) (f2 (f1 x)) x
limit1_in f1 (D_x no_cond x) ?l x
f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:limit1_in (fun x0 : R => f2 (f1 x0))
  (Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1)
  (f2 (f1 x)) x ->
limit1_in (fun x0 : R => f2 (f1 x0))
  (D_x no_cond x) (f2 (f1 x)) x
limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x)) ?l
f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
limit1_in (fun x0 : R => f2 (f1 x0))
  (Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1)
  (f2 (f1 x)) x ->
limit1_in (fun x0 : R => f2 (f1 x0)) (D_x no_cond x)
  (f2 (f1 x)) x
  apply H.f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:limit1_in (fun x0 : R => f2 (f1 x0))
  (Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1)
  (f2 (f1 x)) x ->
limit1_in (fun x0 : R => f2 (f1 x0))
  (D_x no_cond x) (f2 (f1 x)) x
limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x)) (f1 x)
f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
limit1_in (fun x0 : R => f2 (f1 x0))
  (Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1)
  (f2 (f1 x)) x ->
limit1_in (fun x0 : R => f2 (f1 x0)) (D_x no_cond x)
  (f2 (f1 x)) x
  apply H0.f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
limit1_in (fun x0 : R => f2 (f1 x0))
  (Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1)
  (f2 (f1 x)) x ->
limit1_in (fun x0 : R => f2 (f1 x0)) (D_x no_cond x)
  (f2 (f1 x)) x
  unfold limit1_in; unfold limit_in; unfold dist;
    simpl; unfold R_dist; intros.f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x0 /\
   Rabs (x0 - x) < alp ->
   Rabs (f2 (f1 x0) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 (f1 x0) - f2 (f1 x)) < eps)
  assert (H3 := H1 eps H2).f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x0 /\
   Rabs (x0 - x) < alp ->
   Rabs (f2 (f1 x0) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x0 /\
   Rabs (x0 - x) < alp ->
   Rabs (f2 (f1 x0) - f2 (f1 x)) < eps)
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f2 (f1 x0) - f2 (f1 x)) < eps)
  elim H3; intros.f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x1 /\
   Rabs (x1 - x) < alp ->
   Rabs (f2 (f1 x1) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x1 /\
   Rabs (x1 - x) < alp ->
   Rabs (f2 (f1 x1) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x1 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x1 /\
 Rabs (x1 - x) < x0 ->
 Rabs (f2 (f1 x1) - f2 (f1 x)) < eps)
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond x x1 /\ Rabs (x1 - x) < alp ->
   Rabs (f2 (f1 x1) - f2 (f1 x)) < eps)
  exists x0.f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x1 /\
   Rabs (x1 - x) < alp ->
   Rabs (f2 (f1 x1) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x1 /\
   Rabs (x1 - x) < alp ->
   Rabs (f2 (f1 x1) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x1 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x1 /\
 Rabs (x1 - x) < x0 ->
 Rabs (f2 (f1 x1) - f2 (f1 x)) < eps)
x0 > 0 /\
(forall x1 : R,
 D_x no_cond x x1 /\ Rabs (x1 - x) < x0 ->
 Rabs (f2 (f1 x1) - f2 (f1 x)) < eps)
  split.f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x1 /\
   Rabs (x1 - x) < alp ->
   Rabs (f2 (f1 x1) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x1 /\
   Rabs (x1 - x) < alp ->
   Rabs (f2 (f1 x1) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x1 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x1 /\
 Rabs (x1 - x) < x0 ->
 Rabs (f2 (f1 x1) - f2 (f1 x)) < eps)
x0 > 0
f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x1 /\
   Rabs (x1 - x) < alp ->
   Rabs (f2 (f1 x1) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x1 /\
   Rabs (x1 - x) < alp ->
   Rabs (f2 (f1 x1) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x1 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x1 /\
 Rabs (x1 - x) < x0 ->
 Rabs (f2 (f1 x1) - f2 (f1 x)) < eps)
forall x1 : R,
D_x no_cond x x1 /\ Rabs (x1 - x) < x0 ->
Rabs (f2 (f1 x1) - f2 (f1 x)) < eps
  elim H4; intros; assumption.f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x1 /\
   Rabs (x1 - x) < alp ->
   Rabs (f2 (f1 x1) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x1 /\
   Rabs (x1 - x) < alp ->
   Rabs (f2 (f1 x1) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x1 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x1 /\
 Rabs (x1 - x) < x0 ->
 Rabs (f2 (f1 x1) - f2 (f1 x)) < eps)
forall x1 : R,
D_x no_cond x x1 /\ Rabs (x1 - x) < x0 ->
Rabs (f2 (f1 x1) - f2 (f1 x)) < eps
  intros; case (Req_dec (f1 x) (f1 x1)); intro.f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x2 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x1:R
H5:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H6:f1 x = f1 x1
Rabs (f2 (f1 x1) - f2 (f1 x)) < eps
f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x2 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x1:R
H5:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H6:f1 x <> f1 x1
Rabs (f2 (f1 x1) - f2 (f1 x)) < eps
  rewrite H6; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0;
    assumption.f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x2 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x1:R
H5:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H6:f1 x <> f1 x1
Rabs (f2 (f1 x1) - f2 (f1 x)) < eps
  elim H4; intros; apply H8.f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x2 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x1:R
H5:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H6:f1 x <> f1 x1
H7:x0 > 0
H8:forall x2 : R,
Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
Rabs (x2 - x) < x0 ->
Rabs (f2 (f1 x2) - f2 (f1 x)) < eps
Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x1 /\
Rabs (x1 - x) < x0
  split.f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x2 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x1:R
H5:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H6:f1 x <> f1 x1
H7:x0 > 0
H8:forall x2 : R,
Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
Rabs (x2 - x) < x0 ->
Rabs (f2 (f1 x2) - f2 (f1 x)) < eps
Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x1
f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x2 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x1:R
H5:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H6:f1 x <> f1 x1
H7:x0 > 0
H8:forall x2 : R,
Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
Rabs (x2 - x) < x0 ->
Rabs (f2 (f1 x2) - f2 (f1 x)) < eps
Rabs (x1 - x) < x0
  unfold Dgf, D_x, no_cond.f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x2 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x1:R
H5:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H6:f1 x <> f1 x1
H7:x0 > 0
H8:forall x2 : R,
Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
Rabs (x2 - x) < x0 ->
Rabs (f2 (f1 x2) - f2 (f1 x)) < eps
(True /\ x <> x1) /\ True /\ f1 x <> f1 x1
f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x2 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x1:R
H5:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H6:f1 x <> f1 x1
H7:x0 > 0
H8:forall x2 : R,
Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
Rabs (x2 - x) < x0 ->
Rabs (f2 (f1 x2) - f2 (f1 x)) < eps
Rabs (x1 - x) < x0
  split.f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x2 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x1:R
H5:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H6:f1 x <> f1 x1
H7:x0 > 0
H8:forall x2 : R,
Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
Rabs (x2 - x) < x0 ->
Rabs (f2 (f1 x2) - f2 (f1 x)) < eps
True /\ x <> x1
f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x2 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x1:R
H5:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H6:f1 x <> f1 x1
H7:x0 > 0
H8:forall x2 : R,
Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
Rabs (x2 - x) < x0 ->
Rabs (f2 (f1 x2) - f2 (f1 x)) < eps
True /\ f1 x <> f1 x1
f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x2 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x1:R
H5:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H6:f1 x <> f1 x1
H7:x0 > 0
H8:forall x2 : R,
Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
Rabs (x2 - x) < x0 ->
Rabs (f2 (f1 x2) - f2 (f1 x)) < eps
Rabs (x1 - x) < x0
  split.f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x2 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x1:R
H5:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H6:f1 x <> f1 x1
H7:x0 > 0
H8:forall x2 : R,
Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
Rabs (x2 - x) < x0 ->
Rabs (f2 (f1 x2) - f2 (f1 x)) < eps
True
f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x2 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x1:R
H5:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H6:f1 x <> f1 x1
H7:x0 > 0
H8:forall x2 : R,
Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
Rabs (x2 - x) < x0 ->
Rabs (f2 (f1 x2) - f2 (f1 x)) < eps
x <> x1
f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x2 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x1:R
H5:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H6:f1 x <> f1 x1
H7:x0 > 0
H8:forall x2 : R,
Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
Rabs (x2 - x) < x0 ->
Rabs (f2 (f1 x2) - f2 (f1 x)) < eps
True /\ f1 x <> f1 x1
f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x2 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x1:R
H5:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H6:f1 x <> f1 x1
H7:x0 > 0
H8:forall x2 : R,
Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
Rabs (x2 - x) < x0 ->
Rabs (f2 (f1 x2) - f2 (f1 x)) < eps
Rabs (x1 - x) < x0
  trivial.f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x2 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x1:R
H5:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H6:f1 x <> f1 x1
H7:x0 > 0
H8:forall x2 : R,
Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
Rabs (x2 - x) < x0 ->
Rabs (f2 (f1 x2) - f2 (f1 x)) < eps
x <> x1
f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x2 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x1:R
H5:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H6:f1 x <> f1 x1
H7:x0 > 0
H8:forall x2 : R,
Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
Rabs (x2 - x) < x0 ->
Rabs (f2 (f1 x2) - f2 (f1 x)) < eps
True /\ f1 x <> f1 x1
f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x2 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x1:R
H5:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H6:f1 x <> f1 x1
H7:x0 > 0
H8:forall x2 : R,
Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
Rabs (x2 - x) < x0 ->
Rabs (f2 (f1 x2) - f2 (f1 x)) < eps
Rabs (x1 - x) < x0
  elim H5; unfold D_x, no_cond; intros.f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x2 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x1:R
H5:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H6:f1 x <> f1 x1
H7:x0 > 0
H8:forall x2 : R,
Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
Rabs (x2 - x) < x0 ->
Rabs (f2 (f1 x2) - f2 (f1 x)) < eps
H9:True /\ x <> x1
H10:Rabs (x1 - x) < x0
x <> x1
f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x2 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x1:R
H5:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H6:f1 x <> f1 x1
H7:x0 > 0
H8:forall x2 : R,
Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
Rabs (x2 - x) < x0 ->
Rabs (f2 (f1 x2) - f2 (f1 x)) < eps
True /\ f1 x <> f1 x1
f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x2 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x1:R
H5:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H6:f1 x <> f1 x1
H7:x0 > 0
H8:forall x2 : R,
Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
Rabs (x2 - x) < x0 ->
Rabs (f2 (f1 x2) - f2 (f1 x)) < eps
Rabs (x1 - x) < x0
  elim H9; intros; assumption.f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x2 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x1:R
H5:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H6:f1 x <> f1 x1
H7:x0 > 0
H8:forall x2 : R,
Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
Rabs (x2 - x) < x0 ->
Rabs (f2 (f1 x2) - f2 (f1 x)) < eps
True /\ f1 x <> f1 x1
f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x2 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x1:R
H5:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H6:f1 x <> f1 x1
H7:x0 > 0
H8:forall x2 : R,
Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
Rabs (x2 - x) < x0 ->
Rabs (f2 (f1 x2) - f2 (f1 x)) < eps
Rabs (x1 - x) < x0
  split.f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x2 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x1:R
H5:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H6:f1 x <> f1 x1
H7:x0 > 0
H8:forall x2 : R,
Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
Rabs (x2 - x) < x0 ->
Rabs (f2 (f1 x2) - f2 (f1 x)) < eps
True
f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x2 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x1:R
H5:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H6:f1 x <> f1 x1
H7:x0 > 0
H8:forall x2 : R,
Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
Rabs (x2 - x) < x0 ->
Rabs (f2 (f1 x2) - f2 (f1 x)) < eps
f1 x <> f1 x1
f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x2 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x1:R
H5:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H6:f1 x <> f1 x1
H7:x0 > 0
H8:forall x2 : R,
Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
Rabs (x2 - x) < x0 ->
Rabs (f2 (f1 x2) - f2 (f1 x)) < eps
Rabs (x1 - x) < x0
  trivial.f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x2 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x1:R
H5:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H6:f1 x <> f1 x1
H7:x0 > 0
H8:forall x2 : R,
Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
Rabs (x2 - x) < x0 ->
Rabs (f2 (f1 x2) - f2 (f1 x)) < eps
f1 x <> f1 x1
f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x2 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x1:R
H5:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H6:f1 x <> f1 x1
H7:x0 > 0
H8:forall x2 : R,
Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
Rabs (x2 - x) < x0 ->
Rabs (f2 (f1 x2) - f2 (f1 x)) < eps
Rabs (x1 - x) < x0
  assumption.f1, f2:R -> R
x:R
H:limit1_in f1 (D_x no_cond x) (f1 x) x
H0:limit1_in f2 (D_x no_cond (f1 x)) (f2 (f1 x))
  (f1 x)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps0)
eps:R
H2:eps > 0
H3:exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
   Rabs (x2 - x) < alp ->
   Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x0:R
H4:x0 > 0 /\
(forall x2 : R,
 Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs (f2 (f1 x2) - f2 (f1 x)) < eps)
x1:R
H5:D_x no_cond x x1 /\ Rabs (x1 - x) < x0
H6:f1 x <> f1 x1
H7:x0 > 0
H8:forall x2 : R,
Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1 x2 /\
Rabs (x2 - x) < x0 ->
Rabs (f2 (f1 x2) - f2 (f1 x)) < eps
Rabs (x1 - x) < x0
  elim H5; intros; assumption.
Qed.

(**********)
Lemma continuity_plus :
  forall f1 f2, continuity f1 -> continuity f2 -> continuity (f1 + f2).forall f1 f2 : R -> R,
continuity f1 -> continuity f2 -> continuity (f1 + f2)
Proof.forall f1 f2 : R -> R,
continuity f1 -> continuity f2 -> continuity (f1 + f2)
  unfold continuity; intros;
    apply (continuity_pt_plus f1 f2 x (H x) (H0 x)).
Qed.

Lemma continuity_opp : forall f, continuity f -> continuity (- f).forall f : R -> R, continuity f -> continuity (- f)
Proof.forall f : R -> R, continuity f -> continuity (- f)
  unfold continuity; intros; apply (continuity_pt_opp f x (H x)).
Qed.

Lemma continuity_minus :
  forall f1 f2, continuity f1 -> continuity f2 -> continuity (f1 - f2).forall f1 f2 : R -> R,
continuity f1 -> continuity f2 -> continuity (f1 - f2)
Proof.forall f1 f2 : R -> R,
continuity f1 -> continuity f2 -> continuity (f1 - f2)
  unfold continuity; intros;
    apply (continuity_pt_minus f1 f2 x (H x) (H0 x)).
Qed.

Lemma continuity_mult :
  forall f1 f2, continuity f1 -> continuity f2 -> continuity (f1 * f2).forall f1 f2 : R -> R,
continuity f1 -> continuity f2 -> continuity (f1 * f2)
Proof.forall f1 f2 : R -> R,
continuity f1 -> continuity f2 -> continuity (f1 * f2)
  unfold continuity; intros;
    apply (continuity_pt_mult f1 f2 x (H x) (H0 x)).
Qed.

Lemma continuity_const : forall f, constant f -> continuity f.forall f : R -> R, constant f -> continuity f
Proof.forall f : R -> R, constant f -> continuity f
  unfold continuity; intros; apply (continuity_pt_const f x H).
Qed.

Lemma continuity_scal :
  forall f (a:R), continuity f -> continuity (mult_real_fct a f).forall (f : R -> R) (a : R),
continuity f -> continuity (mult_real_fct a f)
Proof.forall (f : R -> R) (a : R),
continuity f -> continuity (mult_real_fct a f)
  unfold continuity; intros; apply (continuity_pt_scal f a x (H x)).
Qed.

Lemma continuity_inv :
  forall f, continuity f -> (forall x:R, f x <> 0) -> continuity (/ f).forall f : R -> R,
continuity f ->
(forall x : R, f x <> 0) -> continuity (/ f)
Proof.forall f : R -> R,
continuity f ->
(forall x : R, f x <> 0) -> continuity (/ f)
  unfold continuity; intros; apply (continuity_pt_inv f x (H x) (H0 x)).
Qed.

Lemma continuity_div :
  forall f1 f2,
    continuity f1 ->
    continuity f2 -> (forall x:R, f2 x <> 0) -> continuity (f1 / f2).forall f1 f2 : R -> R,
continuity f1 ->
continuity f2 ->
(forall x : R, f2 x <> 0) -> continuity (f1 / f2)
Proof.forall f1 f2 : R -> R,
continuity f1 ->
continuity f2 ->
(forall x : R, f2 x <> 0) -> continuity (f1 / f2)
  unfold continuity; intros;
    apply (continuity_pt_div f1 f2 x (H x) (H0 x) (H1 x)).
Qed.

Lemma continuity_comp :
  forall f1 f2, continuity f1 -> continuity f2 -> continuity (f2 o f1).forall f1 f2 : R -> R,
continuity f1 -> continuity f2 -> continuity (f2 o f1)
Proof.forall f1 f2 : R -> R,
continuity f1 -> continuity f2 -> continuity (f2 o f1)
  unfold continuity; intros.f1, f2:R -> R
H:forall x0 : R, continuity_pt f1 x0
H0:forall x0 : R, continuity_pt f2 x0
x:R
continuity_pt (f2 o f1) x
  apply (continuity_pt_comp f1 f2 x (H x) (H0 (f1 x))).
Qed.


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

Derivative's definition using Landau's kernel

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

Definition derivable_pt_lim f (x l:R) : Prop :=
  forall eps:R,
    0 < eps ->
    exists delta : posreal,
      (forall h:R,
        h <> 0 -> Rabs h < delta -> Rabs ((f (x + h) - f x) / h - l) < eps).

Definition derivable_pt_abs f (x l:R) : Prop := derivable_pt_lim f x l.

Definition derivable_pt f (x:R) := { l:R | derivable_pt_abs f x l }.
Definition derivable f := forall x:R, derivable_pt f x.

Definition derive_pt f (x:R) (pr:derivable_pt f x) := proj1_sig pr.
Definition derive f (pr:derivable f) (x:R) := derive_pt f x (pr x).

Arguments derivable_pt_lim f%F x%R l.
Arguments derivable_pt_abs f%F (x l)%R.
Arguments derivable_pt f%F x%R.
Arguments derivable f%F.
Arguments derive_pt f%F x%R pr.
Arguments derive f%F pr x.

Definition antiderivative f (g:R -> R) (a b:R) : Prop :=
  (forall x:R,
    a <= x <= b ->  exists pr : derivable_pt g x, f x = derive_pt g x pr) /\
  a <= b.
(**************************************)

Class of differential functions

(**************************************)
Record Differential : Type := mkDifferential
  {d1 :> R -> R; cond_diff : derivable d1}.

Record Differential_D2 : Type := mkDifferential_D2
  {d2 :> R -> R;
    cond_D1 : derivable d2;
    cond_D2 : derivable (derive d2 cond_D1)}.

(**********)
Lemma uniqueness_step1 :
  forall f (x l1 l2:R),
    limit1_in (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) l1 0 ->
    limit1_in (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) l2 0 ->
    l1 = l2.forall (f : R -> R) (x l1 l2 : R),
limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l1 0 ->
limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l2 0 -> l1 = l2
Proof.forall (f : R -> R) (x l1 l2 : R),
limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l1 0 ->
limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l2 0 -> l1 = l2
  intros;
    apply
      (single_limit (fun h:R => (f (x + h) - f x) / h) (
        fun h:R => h <> 0) l1 l2 0); try assumption.f:R -> R
x, l1, l2:R
H:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l1 0
H0:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l2 0
adhDa (fun h : R => h <> 0) 0
  unfold adhDa; intros; exists (alp / 2).f:R -> R
x, l1, l2:R
H:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l1 0
H0:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l2 0
alp:R
H1:alp > 0
alp / 2 <> 0 /\ R_dist (alp / 2) 0 < alp
  split.f:R -> R
x, l1, l2:R
H:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l1 0
H0:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l2 0
alp:R
H1:alp > 0
alp / 2 <> 0
f:R -> R
x, l1, l2:R
H:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l1 0
H0:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l2 0
alp:R
H1:alp > 0
R_dist (alp / 2) 0 < alp
  unfold Rdiv; apply prod_neq_R0.f:R -> R
x, l1, l2:R
H:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l1 0
H0:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l2 0
alp:R
H1:alp > 0
alp <> 0
f:R -> R
x, l1, l2:R
H:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l1 0
H0:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l2 0
alp:R
H1:alp > 0
/ 2 <> 0
f:R -> R
x, l1, l2:R
H:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l1 0
H0:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l2 0
alp:R
H1:alp > 0
R_dist (alp / 2) 0 < alp
  red; intro; rewrite H2 in H1; elim (Rlt_irrefl _ H1).f:R -> R
x, l1, l2:R
H:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l1 0
H0:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l2 0
alp:R
H1:alp > 0
/ 2 <> 0
f:R -> R
x, l1, l2:R
H:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l1 0
H0:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l2 0
alp:R
H1:alp > 0
R_dist (alp / 2) 0 < alp
  apply Rinv_neq_0_compat; discrR.f:R -> R
x, l1, l2:R
H:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l1 0
H0:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l2 0
alp:R
H1:alp > 0
R_dist (alp / 2) 0 < alp
  unfold R_dist; unfold Rminus; rewrite Ropp_0;
    rewrite Rplus_0_r; unfold Rdiv; rewrite Rabs_mult.f:R -> R
x, l1, l2:R
H:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l1 0
H0:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l2 0
alp:R
H1:alp > 0
Rabs alp * Rabs (/ 2) < alp
  replace (Rabs (/ 2)) with (/ 2).f:R -> R
x, l1, l2:R
H:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l1 0
H0:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l2 0
alp:R
H1:alp > 0
Rabs alp * / 2 < alp
f:R -> R
x, l1, l2:R
H:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l1 0
H0:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l2 0
alp:R
H1:alp > 0
/ 2 = Rabs (/ 2)
  replace (Rabs alp) with alp.f:R -> R
x, l1, l2:R
H:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l1 0
H0:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l2 0
alp:R
H1:alp > 0
alp * / 2 < alp
f:R -> R
x, l1, l2:R
H:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l1 0
H0:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l2 0
alp:R
H1:alp > 0
alp = Rabs alp
f:R -> R
x, l1, l2:R
H:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l1 0
H0:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l2 0
alp:R
H1:alp > 0
/ 2 = Rabs (/ 2)
  apply Rmult_lt_reg_l with 2.f:R -> R
x, l1, l2:R
H:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l1 0
H0:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l2 0
alp:R
H1:alp > 0
0 < 2
f:R -> R
x, l1, l2:R
H:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l1 0
H0:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l2 0
alp:R
H1:alp > 0
2 * (alp * / 2) < 2 * alp
f:R -> R
x, l1, l2:R
H:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l1 0
H0:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l2 0
alp:R
H1:alp > 0
alp = Rabs alp
f:R -> R
x, l1, l2:R
H:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l1 0
H0:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l2 0
alp:R
H1:alp > 0
/ 2 = Rabs (/ 2)
  prove_sup0.f:R -> R
x, l1, l2:R
H:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l1 0
H0:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l2 0
alp:R
H1:alp > 0
2 * (alp * / 2) < 2 * alp
f:R -> R
x, l1, l2:R
H:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l1 0
H0:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l2 0
alp:R
H1:alp > 0
alp = Rabs alp
f:R -> R
x, l1, l2:R
H:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l1 0
H0:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l2 0
alp:R
H1:alp > 0
/ 2 = Rabs (/ 2)
  rewrite (Rmult_comm 2); rewrite Rmult_assoc; rewrite <- Rinv_l_sym;
    [ idtac | discrR ]; rewrite Rmult_1_r; rewrite double;
      pattern alp at 1; replace alp with (alp + 0);
        [ idtac | ring ]; apply Rplus_lt_compat_l; assumption.f:R -> R
x, l1, l2:R
H:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l1 0
H0:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l2 0
alp:R
H1:alp > 0
alp = Rabs alp
f:R -> R
x, l1, l2:R
H:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l1 0
H0:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l2 0
alp:R
H1:alp > 0
/ 2 = Rabs (/ 2)
  symmetry ; apply Rabs_right; left; assumption.f:R -> R
x, l1, l2:R
H:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l1 0
H0:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l2 0
alp:R
H1:alp > 0
/ 2 = Rabs (/ 2)
  symmetry ; apply Rabs_right; left; change (0 < / 2);
    apply Rinv_0_lt_compat; prove_sup0.
Qed.

Lemma uniqueness_step2 :
  forall f (x l:R),
    derivable_pt_lim f x l ->
    limit1_in (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) l 0.forall (f : R -> R) (x l : R),
derivable_pt_lim f x l ->
limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l 0
Proof.forall (f : R -> R) (x l : R),
derivable_pt_lim f x l ->
limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l 0
  unfold derivable_pt_lim; intros; unfold limit1_in;
    unfold limit_in; intros.f:R -> R
x, l:R
H:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - l) < eps0
eps:R
H0:eps > 0
exists alp : R,
  alp > 0 /\
  (forall x0 : Base R_met,
   x0 <> 0 /\ dist R_met x0 0 < alp ->
   dist R_met ((f (x + x0) - f x) / x0) l < eps)
  assert (H1 := H eps H0).f:R -> R
x, l:R
H:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - l) < eps0
eps:R
H0:eps > 0
H1:exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - l) < eps
exists alp : R,
  alp > 0 /\
  (forall x0 : Base R_met,
   x0 <> 0 /\ dist R_met x0 0 < alp ->
   dist R_met ((f (x + x0) - f x) / x0) l < eps)
  elim H1; intros.f:R -> R
x, l:R
H:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - l) < eps0
eps:R
H0:eps > 0
H1:exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - l) < eps
x0:posreal
H2:forall h : R,
h <> 0 ->
Rabs h < x0 ->
Rabs ((f (x + h) - f x) / h - l) < eps
exists alp : R,
  alp > 0 /\
  (forall x1 : Base R_met,
   x1 <> 0 /\ dist R_met x1 0 < alp ->
   dist R_met ((f (x + x1) - f x) / x1) l < eps)
  exists (pos x0).f:R -> R
x, l:R
H:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - l) < eps0
eps:R
H0:eps > 0
H1:exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - l) < eps
x0:posreal
H2:forall h : R,
h <> 0 ->
Rabs h < x0 ->
Rabs ((f (x + h) - f x) / h - l) < eps
x0 > 0 /\
(forall x1 : Base R_met,
 x1 <> 0 /\ dist R_met x1 0 < x0 ->
 dist R_met ((f (x + x1) - f x) / x1) l < eps)
  split.f:R -> R
x, l:R
H:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - l) < eps0
eps:R
H0:eps > 0
H1:exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - l) < eps
x0:posreal
H2:forall h : R,
h <> 0 ->
Rabs h < x0 ->
Rabs ((f (x + h) - f x) / h - l) < eps
x0 > 0
f:R -> R
x, l:R
H:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - l) < eps0
eps:R
H0:eps > 0
H1:exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - l) < eps
x0:posreal
H2:forall h : R,
h <> 0 ->
Rabs h < x0 ->
Rabs ((f (x + h) - f x) / h - l) < eps
forall x1 : Base R_met,
x1 <> 0 /\ dist R_met x1 0 < x0 ->
dist R_met ((f (x + x1) - f x) / x1) l < eps
  apply (cond_pos x0).f:R -> R
x, l:R
H:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - l) < eps0
eps:R
H0:eps > 0
H1:exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - l) < eps
x0:posreal
H2:forall h : R,
h <> 0 ->
Rabs h < x0 ->
Rabs ((f (x + h) - f x) / h - l) < eps
forall x1 : Base R_met,
x1 <> 0 /\ dist R_met x1 0 < x0 ->
dist R_met ((f (x + x1) - f x) / x1) l < eps
  simpl; unfold R_dist; intros.f:R -> R
x, l:R
H:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - l) < eps0
eps:R
H0:eps > 0
H1:exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - l) < eps
x0:posreal
H2:forall h : R,
h <> 0 ->
Rabs h < x0 ->
Rabs ((f (x + h) - f x) / h - l) < eps
x1:R
H3:x1 <> 0 /\ Rabs (x1 - 0) < x0
Rabs ((f (x + x1) - f x) / x1 - l) < eps
  elim H3; intros.f:R -> R
x, l:R
H:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - l) < eps0
eps:R
H0:eps > 0
H1:exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - l) < eps
x0:posreal
H2:forall h : R,
h <> 0 ->
Rabs h < x0 ->
Rabs ((f (x + h) - f x) / h - l) < eps
x1:R
H3:x1 <> 0 /\ Rabs (x1 - 0) < x0
H4:x1 <> 0
H5:Rabs (x1 - 0) < x0
Rabs ((f (x + x1) - f x) / x1 - l) < eps
  apply H2;
    [ assumption
      | unfold Rminus in H5; rewrite Ropp_0 in H5; rewrite Rplus_0_r in H5;
        assumption ].
Qed.

Lemma uniqueness_step3 :
  forall f (x l:R),
    limit1_in (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) l 0 ->
    derivable_pt_lim f x l.forall (f : R -> R) (x l : R),
limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l 0 -> derivable_pt_lim f x l
Proof.forall (f : R -> R) (x l : R),
limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l 0 -> derivable_pt_lim f x l
  unfold limit1_in, derivable_pt_lim; unfold limit_in;
    unfold dist; simpl; intros.f:R -> R
x, l:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   x0 <> 0 /\ R_dist x0 0 < alp ->
   R_dist ((f (x + x0) - f x) / x0) l < eps0)
eps:R
H0:0 < eps
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - l) < eps
  elim (H eps H0).f:R -> R
x, l:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   x0 <> 0 /\ R_dist x0 0 < alp ->
   R_dist ((f (x + x0) - f x) / x0) l < eps0)
eps:R
H0:0 < eps
forall x0 : R,
x0 > 0 /\
(forall x1 : R,
 x1 <> 0 /\ R_dist x1 0 < x0 ->
 R_dist ((f (x + x1) - f x) / x1) l < eps) ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - l) < eps
  intros; elim H1; intros.f:R -> R
x, l:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   x1 <> 0 /\ R_dist x1 0 < alp ->
   R_dist ((f (x + x1) - f x) / x1) l < eps0)
eps:R
H0:0 < eps
x0:R
H1:x0 > 0 /\
(forall x1 : R,
 x1 <> 0 /\ R_dist x1 0 < x0 ->
 R_dist ((f (x + x1) - f x) / x1) l < eps)
H2:x0 > 0
H3:forall x1 : R,
x1 <> 0 /\ R_dist x1 0 < x0 ->
R_dist ((f (x + x1) - f x) / x1) l < eps
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - l) < eps
  exists (mkposreal x0 H2).f:R -> R
x, l:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   x1 <> 0 /\ R_dist x1 0 < alp ->
   R_dist ((f (x + x1) - f x) / x1) l < eps0)
eps:R
H0:0 < eps
x0:R
H1:x0 > 0 /\
(forall x1 : R,
 x1 <> 0 /\ R_dist x1 0 < x0 ->
 R_dist ((f (x + x1) - f x) / x1) l < eps)
H2:x0 > 0
H3:forall x1 : R,
x1 <> 0 /\ R_dist x1 0 < x0 ->
R_dist ((f (x + x1) - f x) / x1) l < eps
forall h : R,
h <> 0 ->
Rabs h < {| pos := x0; cond_pos := H2 |} ->
Rabs ((f (x + h) - f x) / h - l) < eps
  simpl; intros; unfold R_dist in H3; apply (H3 h).f:R -> R
x, l:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   x1 <> 0 /\ R_dist x1 0 < alp ->
   R_dist ((f (x + x1) - f x) / x1) l < eps0)
eps:R
H0:0 < eps
x0:R
H1:x0 > 0 /\
(forall x1 : R,
 x1 <> 0 /\ R_dist x1 0 < x0 ->
 R_dist ((f (x + x1) - f x) / x1) l < eps)
H2:x0 > 0
H3:forall x1 : R,
x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
Rabs ((f (x + x1) - f x) / x1 - l) < eps
h:R
H4:h <> 0
H5:Rabs h < x0
h <> 0 /\ Rabs (h - 0) < x0
  split;
    [ assumption
      | unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; assumption ].
Qed.

Lemma uniqueness_limite :
  forall f (x l1 l2:R),
    derivable_pt_lim f x l1 -> derivable_pt_lim f x l2 -> l1 = l2.forall (f : R -> R) (x l1 l2 : R),
derivable_pt_lim f x l1 ->
derivable_pt_lim f x l2 -> l1 = l2
Proof.forall (f : R -> R) (x l1 l2 : R),
derivable_pt_lim f x l1 ->
derivable_pt_lim f x l2 -> l1 = l2
  intros.f:R -> R
x, l1, l2:R
H:derivable_pt_lim f x l1
H0:derivable_pt_lim f x l2
l1 = l2
  assert (H1 := uniqueness_step2 _ _ _ H).f:R -> R
x, l1, l2:R
H:derivable_pt_lim f x l1
H0:derivable_pt_lim f x l2
H1:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l1 0
l1 = l2
  assert (H2 := uniqueness_step2 _ _ _ H0).f:R -> R
x, l1, l2:R
H:derivable_pt_lim f x l1
H0:derivable_pt_lim f x l2
H1:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l2 0
l1 = l2
  assert (H3 := uniqueness_step1 _ _ _ _ H1 H2).f:R -> R
x, l1, l2:R
H:derivable_pt_lim f x l1
H0:derivable_pt_lim f x l2
H1:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l2 0
H3:l1 = l2
l1 = l2
  assumption.
Qed.

Lemma derive_pt_eq :
  forall f (x l:R) (pr:derivable_pt f x),
    derive_pt f x pr = l <-> derivable_pt_lim f x l.forall (f : R -> R) (x l : R) (pr : derivable_pt f x),
derive_pt f x pr = l <-> derivable_pt_lim f x l
Proof.forall (f : R -> R) (x l : R) (pr : derivable_pt f x),
derive_pt f x pr = l <-> derivable_pt_lim f x l
  intros; split.f:R -> R
x, l:R
pr:derivable_pt f x
derive_pt f x pr = l -> derivable_pt_lim f x l
f:R -> R
x, l:R
pr:derivable_pt f x
derivable_pt_lim f x l -> derive_pt f x pr = l
  intro; assert (H1 := proj2_sig pr); unfold derive_pt in H; rewrite H in H1;
    assumption.f:R -> R
x, l:R
pr:derivable_pt f x
derivable_pt_lim f x l -> derive_pt f x pr = l
  intro; assert (H1 := proj2_sig pr); unfold derivable_pt_abs in H1.f:R -> R
x, l:R
pr:derivable_pt f x
H:derivable_pt_lim f x l
H1:derivable_pt_lim f x (proj1_sig pr)
derive_pt f x pr = l
  assert (H2 := uniqueness_limite _ _ _ _ H H1).f:R -> R
x, l:R
pr:derivable_pt f x
H:derivable_pt_lim f x l
H1:derivable_pt_lim f x (proj1_sig pr)
H2:l = proj1_sig pr
derive_pt f x pr = l
  unfold derive_pt; unfold derivable_pt_abs.f:R -> R
x, l:R
pr:derivable_pt f x
H:derivable_pt_lim f x l
H1:derivable_pt_lim f x (proj1_sig pr)
H2:l = proj1_sig pr
proj1_sig pr = l
  symmetry ; assumption.
Qed.

(**********)
Lemma derive_pt_eq_0 :
  forall f (x l:R) (pr:derivable_pt f x),
    derivable_pt_lim f x l -> derive_pt f x pr = l.forall (f : R -> R) (x l : R) (pr : derivable_pt f x),
derivable_pt_lim f x l -> derive_pt f x pr = l
Proof.forall (f : R -> R) (x l : R) (pr : derivable_pt f x),
derivable_pt_lim f x l -> derive_pt f x pr = l
  intros; elim (derive_pt_eq f x l pr); intros.f:R -> R
x, l:R
pr:derivable_pt f x
H:derivable_pt_lim f x l
H0:derive_pt f x pr = l -> derivable_pt_lim f x l
H1:derivable_pt_lim f x l -> derive_pt f x pr = l
derive_pt f x pr = l
  apply (H1 H).
Qed.

(**********)
Lemma derive_pt_eq_1 :
  forall f (x l:R) (pr:derivable_pt f x),
    derive_pt f x pr = l -> derivable_pt_lim f x l.forall (f : R -> R) (x l : R) (pr : derivable_pt f x),
derive_pt f x pr = l -> derivable_pt_lim f x l
Proof.forall (f : R -> R) (x l : R) (pr : derivable_pt f x),
derive_pt f x pr = l -> derivable_pt_lim f x l
  intros; elim (derive_pt_eq f x l pr); intros.f:R -> R
x, l:R
pr:derivable_pt f x
H:derive_pt f x pr = l
H0:derive_pt f x pr = l -> derivable_pt_lim f x l
H1:derivable_pt_lim f x l -> derive_pt f x pr = l
derivable_pt_lim f x l
  apply (H0 H).
Qed.


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

Equivalence of this definition with the one using limit concept

(**********************************************************************)
Lemma derive_pt_D_in :
  forall f (df:R -> R) (x:R) (pr:derivable_pt f x),
    D_in f df no_cond x <-> derive_pt f x pr = df x.forall (f df : R -> R) (x : R) (pr : derivable_pt f x),
D_in f df no_cond x <-> derive_pt f x pr = df x
Proof.forall (f df : R -> R) (x : R) (pr : derivable_pt f x),
D_in f df no_cond x <-> derive_pt f x pr = df x
  intros; split.f, df:R -> R
x:R
pr:derivable_pt f x
D_in f df no_cond x -> derive_pt f x pr = df x
f, df:R -> R
x:R
pr:derivable_pt f x
derive_pt f x pr = df x -> D_in f df no_cond x
  unfold D_in; unfold limit1_in; unfold limit_in;
    simpl; unfold R_dist; intros.f, df:R -> R
x:R
pr:derivable_pt f x
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs ((f x0 - f x) / (x0 - x) - df x) < eps)
derive_pt f x pr = df x
f, df:R -> R
x:R
pr:derivable_pt f x
derive_pt f x pr = df x -> D_in f df no_cond x
  apply derive_pt_eq_0.f, df:R -> R
x:R
pr:derivable_pt f x
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs ((f x0 - f x) / (x0 - x) - df x) < eps)
derivable_pt_lim f x (df x)
f, df:R -> R
x:R
pr:derivable_pt f x
derive_pt f x pr = df x -> D_in f df no_cond x
  unfold derivable_pt_lim.f, df:R -> R
x:R
pr:derivable_pt f x
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs ((f x0 - f x) / (x0 - x) - df x) < eps)
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - df x) < eps
f, df:R -> R
x:R
pr:derivable_pt f x
derive_pt f x pr = df x -> D_in f df no_cond x
  intros; elim (H eps H0); intros alpha H1; elim H1; intros;
    exists (mkposreal alpha H2); intros; generalize (H3 (x + h));
      intro; cut (x + h - x = h);
        [ intro; cut (D_x no_cond x (x + h) /\ Rabs (x + h - x) < alpha);
          [ intro; generalize (H6 H8); rewrite H7; intro; assumption
            | split;
              [ unfold D_x; split;
                [ unfold no_cond; trivial
                  | apply Rminus_not_eq_right; rewrite H7; assumption ]
                | rewrite H7; assumption ] ]
          | ring ].f, df:R -> R
x:R
pr:derivable_pt f x
derive_pt f x pr = df x -> D_in f df no_cond x
  intro.f, df:R -> R
x:R
pr:derivable_pt f x
H:derive_pt f x pr = df x
D_in f df no_cond x
  assert (H0 := derive_pt_eq_1 f x (df x) pr H).f, df:R -> R
x:R
pr:derivable_pt f x
H:derive_pt f x pr = df x
H0:derivable_pt_lim f x (df x)
D_in f df no_cond x
  unfold D_in; unfold limit1_in; unfold limit_in;
    unfold dist; simpl; unfold R_dist;
      intros.f, df:R -> R
x:R
pr:derivable_pt f x
H:derive_pt f x pr = df x
H0:derivable_pt_lim f x (df x)
eps:R
H1:eps > 0
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs ((f x0 - f x) / (x0 - x) - df x) < eps)
  elim (H0 eps H1); intros alpha H2; exists (pos alpha); split.f, df:R -> R
x:R
pr:derivable_pt f x
H:derive_pt f x pr = df x
H0:derivable_pt_lim f x (df x)
eps:R
H1:eps > 0
alpha:posreal
H2:forall h : R,
h <> 0 ->
Rabs h < alpha ->
Rabs ((f (x + h) - f x) / h - df x) < eps
alpha > 0
f, df:R -> R
x:R
pr:derivable_pt f x
H:derive_pt f x pr = df x
H0:derivable_pt_lim f x (df x)
eps:R
H1:eps > 0
alpha:posreal
H2:forall h : R,
h <> 0 ->
Rabs h < alpha ->
Rabs ((f (x + h) - f x) / h - df x) < eps
forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alpha ->
Rabs ((f x0 - f x) / (x0 - x) - df x) < eps
  apply (cond_pos alpha).f, df:R -> R
x:R
pr:derivable_pt f x
H:derive_pt f x pr = df x
H0:derivable_pt_lim f x (df x)
eps:R
H1:eps > 0
alpha:posreal
H2:forall h : R,
h <> 0 ->
Rabs h < alpha ->
Rabs ((f (x + h) - f x) / h - df x) < eps
forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alpha ->
Rabs ((f x0 - f x) / (x0 - x) - df x) < eps
  intros; elim H3; intros; unfold D_x in H4; elim H4; intros; cut (x0 - x <> 0).f, df:R -> R
x:R
pr:derivable_pt f x
H:derive_pt f x pr = df x
H0:derivable_pt_lim f x (df x)
eps:R
H1:eps > 0
alpha:posreal
H2:forall h : R,
h <> 0 ->
Rabs h < alpha ->
Rabs ((f (x + h) - f x) / h - df x) < eps
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < alpha
H4:no_cond x0 /\ x <> x0
H5:Rabs (x0 - x) < alpha
H6:no_cond x0
H7:x <> x0
x0 - x <> 0 ->
Rabs ((f x0 - f x) / (x0 - x) - df x) < eps
f, df:R -> R
x:R
pr:derivable_pt f x
H:derive_pt f x pr = df x
H0:derivable_pt_lim f x (df x)
eps:R
H1:eps > 0
alpha:posreal
H2:forall h : R,
h <> 0 ->
Rabs h < alpha ->
Rabs ((f (x + h) - f x) / h - df x) < eps
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < alpha
H4:no_cond x0 /\ x <> x0
H5:Rabs (x0 - x) < alpha
H6:no_cond x0
H7:x <> x0
x0 - x <> 0
  intro; generalize (H2 (x0 - x) H8 H5); replace (x + (x0 - x)) with x0.f, df:R -> R
x:R
pr:derivable_pt f x
H:derive_pt f x pr = df x
H0:derivable_pt_lim f x (df x)
eps:R
H1:eps > 0
alpha:posreal
H2:forall h : R,
h <> 0 ->
Rabs h < alpha ->
Rabs ((f (x + h) - f x) / h - df x) < eps
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < alpha
H4:no_cond x0 /\ x <> x0
H5:Rabs (x0 - x) < alpha
H6:no_cond x0
H7:x <> x0
H8:x0 - x <> 0
Rabs ((f x0 - f x) / (x0 - x) - df x) < eps ->
Rabs ((f x0 - f x) / (x0 - x) - df x) < eps
f, df:R -> R
x:R
pr:derivable_pt f x
H:derive_pt f x pr = df x
H0:derivable_pt_lim f x (df x)
eps:R
H1:eps > 0
alpha:posreal
H2:forall h : R,
h <> 0 ->
Rabs h < alpha ->
Rabs ((f (x + h) - f x) / h - df x) < eps
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < alpha
H4:no_cond x0 /\ x <> x0
H5:Rabs (x0 - x) < alpha
H6:no_cond x0
H7:x <> x0
H8:x0 - x <> 0
x0 = x + (x0 - x)
f, df:R -> R
x:R
pr:derivable_pt f x
H:derive_pt f x pr = df x
H0:derivable_pt_lim f x (df x)
eps:R
H1:eps > 0
alpha:posreal
H2:forall h : R,
h <> 0 ->
Rabs h < alpha ->
Rabs ((f (x + h) - f x) / h - df x) < eps
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < alpha
H4:no_cond x0 /\ x <> x0
H5:Rabs (x0 - x) < alpha
H6:no_cond x0
H7:x <> x0
x0 - x <> 0
  intro; assumption.f, df:R -> R
x:R
pr:derivable_pt f x
H:derive_pt f x pr = df x
H0:derivable_pt_lim f x (df x)
eps:R
H1:eps > 0
alpha:posreal
H2:forall h : R,
h <> 0 ->
Rabs h < alpha ->
Rabs ((f (x + h) - f x) / h - df x) < eps
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < alpha
H4:no_cond x0 /\ x <> x0
H5:Rabs (x0 - x) < alpha
H6:no_cond x0
H7:x <> x0
H8:x0 - x <> 0
x0 = x + (x0 - x)
f, df:R -> R
x:R
pr:derivable_pt f x
H:derive_pt f x pr = df x
H0:derivable_pt_lim f x (df x)
eps:R
H1:eps > 0
alpha:posreal
H2:forall h : R,
h <> 0 ->
Rabs h < alpha ->
Rabs ((f (x + h) - f x) / h - df x) < eps
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < alpha
H4:no_cond x0 /\ x <> x0
H5:Rabs (x0 - x) < alpha
H6:no_cond x0
H7:x <> x0
x0 - x <> 0
  ring.f, df:R -> R
x:R
pr:derivable_pt f x
H:derive_pt f x pr = df x
H0:derivable_pt_lim f x (df x)
eps:R
H1:eps > 0
alpha:posreal
H2:forall h : R,
h <> 0 ->
Rabs h < alpha ->
Rabs ((f (x + h) - f x) / h - df x) < eps
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < alpha
H4:no_cond x0 /\ x <> x0
H5:Rabs (x0 - x) < alpha
H6:no_cond x0
H7:x <> x0
x0 - x <> 0
  auto with real.
Qed.

Lemma derivable_pt_lim_D_in :
  forall f (df:R -> R) (x:R),
    D_in f df no_cond x <-> derivable_pt_lim f x (df x).forall (f df : R -> R) (x : R),
D_in f df no_cond x <-> derivable_pt_lim f x (df x)
Proof.forall (f df : R -> R) (x : R),
D_in f df no_cond x <-> derivable_pt_lim f x (df x)
  intros; split.f, df:R -> R
x:R
D_in f df no_cond x -> derivable_pt_lim f x (df x)
f, df:R -> R
x:R
derivable_pt_lim f x (df x) -> D_in f df no_cond x
  unfold D_in; unfold limit1_in; unfold limit_in;
    simpl; unfold R_dist; intros.f, df:R -> R
x:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs ((f x0 - f x) / (x0 - x) - df x) < eps)
derivable_pt_lim f x (df x)
f, df:R -> R
x:R
derivable_pt_lim f x (df x) -> D_in f df no_cond x
  unfold derivable_pt_lim.f, df:R -> R
x:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs ((f x0 - f x) / (x0 - x) - df x) < eps)
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - df x) < eps
f, df:R -> R
x:R
derivable_pt_lim f x (df x) -> D_in f df no_cond x
  intros; elim (H eps H0); intros alpha H1; elim H1; intros;
    exists (mkposreal alpha H2); intros; generalize (H3 (x + h));
      intro; cut (x + h - x = h);
        [ intro; cut (D_x no_cond x (x + h) /\ Rabs (x + h - x) < alpha);
          [ intro; generalize (H6 H8); rewrite H7; intro; assumption
            | split;
              [ unfold D_x; split;
                [ unfold no_cond; trivial
                  | apply Rminus_not_eq_right; rewrite H7; assumption ]
                | rewrite H7; assumption ] ]
          | ring ].f, df:R -> R
x:R
derivable_pt_lim f x (df x) -> D_in f df no_cond x
  intro.f, df:R -> R
x:R
H:derivable_pt_lim f x (df x)
D_in f df no_cond x
  unfold derivable_pt_lim in H.f, df:R -> R
x:R
H:forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - df x) < eps
D_in f df no_cond x
  unfold D_in; unfold limit1_in; unfold limit_in;
    unfold dist; simpl; unfold R_dist;
      intros.f, df:R -> R
x:R
H:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - df x) < eps0
eps:R
H0:eps > 0
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs ((f x0 - f x) / (x0 - x) - df x) < eps)
  elim (H eps H0); intros alpha H2; exists (pos alpha); split.f, df:R -> R
x:R
H:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - df x) < eps0
eps:R
H0:eps > 0
alpha:posreal
H2:forall h : R,
h <> 0 ->
Rabs h < alpha ->
Rabs ((f (x + h) - f x) / h - df x) < eps
alpha > 0
f, df:R -> R
x:R
H:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - df x) < eps0
eps:R
H0:eps > 0
alpha:posreal
H2:forall h : R,
h <> 0 ->
Rabs h < alpha ->
Rabs ((f (x + h) - f x) / h - df x) < eps
forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alpha ->
Rabs ((f x0 - f x) / (x0 - x) - df x) < eps
  apply (cond_pos alpha).f, df:R -> R
x:R
H:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - df x) < eps0
eps:R
H0:eps > 0
alpha:posreal
H2:forall h : R,
h <> 0 ->
Rabs h < alpha ->
Rabs ((f (x + h) - f x) / h - df x) < eps
forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < alpha ->
Rabs ((f x0 - f x) / (x0 - x) - df x) < eps
  intros.f, df:R -> R
x:R
H:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - df x) < eps0
eps:R
H0:eps > 0
alpha:posreal
H2:forall h : R,
h <> 0 ->
Rabs h < alpha ->
Rabs ((f (x + h) - f x) / h - df x) < eps
x0:R
H1:D_x no_cond x x0 /\ Rabs (x0 - x) < alpha
Rabs ((f x0 - f x) / (x0 - x) - df x) < eps
  elim H1; intros; unfold D_x in H3; elim H3; intros; cut (x0 - x <> 0).f, df:R -> R
x:R
H:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - df x) < eps0
eps:R
H0:eps > 0
alpha:posreal
H2:forall h : R,
h <> 0 ->
Rabs h < alpha ->
Rabs ((f (x + h) - f x) / h - df x) < eps
x0:R
H1:D_x no_cond x x0 /\ Rabs (x0 - x) < alpha
H3:no_cond x0 /\ x <> x0
H4:Rabs (x0 - x) < alpha
H5:no_cond x0
H6:x <> x0
x0 - x <> 0 ->
Rabs ((f x0 - f x) / (x0 - x) - df x) < eps
f, df:R -> R
x:R
H:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - df x) < eps0
eps:R
H0:eps > 0
alpha:posreal
H2:forall h : R,
h <> 0 ->
Rabs h < alpha ->
Rabs ((f (x + h) - f x) / h - df x) < eps
x0:R
H1:D_x no_cond x x0 /\ Rabs (x0 - x) < alpha
H3:no_cond x0 /\ x <> x0
H4:Rabs (x0 - x) < alpha
H5:no_cond x0
H6:x <> x0
x0 - x <> 0
  intro; generalize (H2 (x0 - x) H7 H4); replace (x + (x0 - x)) with x0.f, df:R -> R
x:R
H:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - df x) < eps0
eps:R
H0:eps > 0
alpha:posreal
H2:forall h : R,
h <> 0 ->
Rabs h < alpha ->
Rabs ((f (x + h) - f x) / h - df x) < eps
x0:R
H1:D_x no_cond x x0 /\ Rabs (x0 - x) < alpha
H3:no_cond x0 /\ x <> x0
H4:Rabs (x0 - x) < alpha
H5:no_cond x0
H6:x <> x0
H7:x0 - x <> 0
Rabs ((f x0 - f x) / (x0 - x) - df x) < eps ->
Rabs ((f x0 - f x) / (x0 - x) - df x) < eps
f, df:R -> R
x:R
H:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - df x) < eps0
eps:R
H0:eps > 0
alpha:posreal
H2:forall h : R,
h <> 0 ->
Rabs h < alpha ->
Rabs ((f (x + h) - f x) / h - df x) < eps
x0:R
H1:D_x no_cond x x0 /\ Rabs (x0 - x) < alpha
H3:no_cond x0 /\ x <> x0
H4:Rabs (x0 - x) < alpha
H5:no_cond x0
H6:x <> x0
H7:x0 - x <> 0
x0 = x + (x0 - x)
f, df:R -> R
x:R
H:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - df x) < eps0
eps:R
H0:eps > 0
alpha:posreal
H2:forall h : R,
h <> 0 ->
Rabs h < alpha ->
Rabs ((f (x + h) - f x) / h - df x) < eps
x0:R
H1:D_x no_cond x x0 /\ Rabs (x0 - x) < alpha
H3:no_cond x0 /\ x <> x0
H4:Rabs (x0 - x) < alpha
H5:no_cond x0
H6:x <> x0
x0 - x <> 0
  intro; assumption.f, df:R -> R
x:R
H:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - df x) < eps0
eps:R
H0:eps > 0
alpha:posreal
H2:forall h : R,
h <> 0 ->
Rabs h < alpha ->
Rabs ((f (x + h) - f x) / h - df x) < eps
x0:R
H1:D_x no_cond x x0 /\ Rabs (x0 - x) < alpha
H3:no_cond x0 /\ x <> x0
H4:Rabs (x0 - x) < alpha
H5:no_cond x0
H6:x <> x0
H7:x0 - x <> 0
x0 = x + (x0 - x)
f, df:R -> R
x:R
H:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - df x) < eps0
eps:R
H0:eps > 0
alpha:posreal
H2:forall h : R,
h <> 0 ->
Rabs h < alpha ->
Rabs ((f (x + h) - f x) / h - df x) < eps
x0:R
H1:D_x no_cond x x0 /\ Rabs (x0 - x) < alpha
H3:no_cond x0 /\ x <> x0
H4:Rabs (x0 - x) < alpha
H5:no_cond x0
H6:x <> x0
x0 - x <> 0
  ring.f, df:R -> R
x:R
H:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((f (x + h) - f x) / h - df x) < eps0
eps:R
H0:eps > 0
alpha:posreal
H2:forall h : R,
h <> 0 ->
Rabs h < alpha ->
Rabs ((f (x + h) - f x) / h - df x) < eps
x0:R
H1:D_x no_cond x x0 /\ Rabs (x0 - x) < alpha
H3:no_cond x0 /\ x <> x0
H4:Rabs (x0 - x) < alpha
H5:no_cond x0
H6:x <> x0
x0 - x <> 0
  auto with real.
Qed.

(* Extensionally equal functions have the same derivative. *)

Lemma derivable_pt_lim_ext : forall f g x l, (forall z, f z = g z) -> 
  derivable_pt_lim f x l -> derivable_pt_lim g x l.forall (f g : R -> R) (x l : R),
(forall z : R, f z = g z) ->
derivable_pt_lim f x l -> derivable_pt_lim g x l
intros f g x l fg df e ep; destruct (df e ep) as [d pd]; exists d; intros h;
rewrite <- !fg; apply pd.
Qed.

(* extensionally equal functions have the same derivative, locally. *)

Lemma derivable_pt_lim_locally_ext : forall f g x a b l, 
  a < x < b ->
  (forall z, a < z < b -> f z = g z) ->
  derivable_pt_lim f x l -> derivable_pt_lim g x l.forall (f g : R -> R) (x a b l : R),
a < x < b ->
(forall z : R, a < z < b -> f z = g z) ->
derivable_pt_lim f x l -> derivable_pt_lim g x l
intros f g x a b l axb fg df e ep.f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((g (x + h) - g x) / h - l) < e
destruct (df e ep) as [d pd].f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h : R,
h <> 0 ->
Rabs h < d ->
Rabs ((f (x + h) - f x) / h - l) < e
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((g (x + h) - g x) / h - l) < e
assert (d'h : 0 < Rmin d (Rmin (b - x) (x - a))).f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h : R,
h <> 0 ->
Rabs h < d ->
Rabs ((f (x + h) - f x) / h - l) < e
0 < Rmin d (Rmin (b - x) (x - a))
f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h : R,
h <> 0 ->
Rabs h < d ->
Rabs ((f (x + h) - f x) / h - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((g (x + h) - g x) / h - l) < e
 apply Rmin_pos;[apply cond_pos | apply Rmin_pos; apply Rlt_Rminus; tauto].f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h : R,
h <> 0 ->
Rabs h < d ->
Rabs ((f (x + h) - f x) / h - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((g (x + h) - g x) / h - l) < e
exists (mkposreal _ d'h); simpl; intros h hn0 cmp.f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h0 : R,
h0 <> 0 ->
Rabs h0 < d ->
Rabs ((f (x + h0) - f x) / h0 - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
h:R
hn0:h <> 0
cmp:Rabs h < Rmin d (Rmin (b - x) (x - a))
Rabs ((g (x + h) - g x) / h - l) < e
rewrite <- !fg;[ |assumption | ].f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h0 : R,
h0 <> 0 ->
Rabs h0 < d ->
Rabs ((f (x + h0) - f x) / h0 - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
h:R
hn0:h <> 0
cmp:Rabs h < Rmin d (Rmin (b - x) (x - a))
Rabs ((f (x + h) - f x) / h - l) < e
f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h0 : R,
h0 <> 0 ->
Rabs h0 < d ->
Rabs ((f (x + h0) - f x) / h0 - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
h:R
hn0:h <> 0
cmp:Rabs h < Rmin d (Rmin (b - x) (x - a))
a < x + h < b
  apply pd;[assumption |].f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h0 : R,
h0 <> 0 ->
Rabs h0 < d ->
Rabs ((f (x + h0) - f x) / h0 - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
h:R
hn0:h <> 0
cmp:Rabs h < Rmin d (Rmin (b - x) (x - a))
Rabs h < d
f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h0 : R,
h0 <> 0 ->
Rabs h0 < d ->
Rabs ((f (x + h0) - f x) / h0 - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
h:R
hn0:h <> 0
cmp:Rabs h < Rmin d (Rmin (b - x) (x - a))
a < x + h < b
 apply Rlt_le_trans with (1 := cmp), Rmin_l.f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h0 : R,
h0 <> 0 ->
Rabs h0 < d ->
Rabs ((f (x + h0) - f x) / h0 - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
h:R
hn0:h <> 0
cmp:Rabs h < Rmin d (Rmin (b - x) (x - a))
a < x + h < b
assert (-h < x - a).f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h0 : R,
h0 <> 0 ->
Rabs h0 < d ->
Rabs ((f (x + h0) - f x) / h0 - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
h:R
hn0:h <> 0
cmp:Rabs h < Rmin d (Rmin (b - x) (x - a))
- h < x - a
f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h0 : R,
h0 <> 0 ->
Rabs h0 < d ->
Rabs ((f (x + h0) - f x) / h0 - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
h:R
hn0:h <> 0
cmp:Rabs h < Rmin d (Rmin (b - x) (x - a))
H:- h < x - a
a < x + h < b
 apply Rle_lt_trans with (1 := Rle_abs _).f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h0 : R,
h0 <> 0 ->
Rabs h0 < d ->
Rabs ((f (x + h0) - f x) / h0 - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
h:R
hn0:h <> 0
cmp:Rabs h < Rmin d (Rmin (b - x) (x - a))
Rabs (- h) < x - a
f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h0 : R,
h0 <> 0 ->
Rabs h0 < d ->
Rabs ((f (x + h0) - f x) / h0 - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
h:R
hn0:h <> 0
cmp:Rabs h < Rmin d (Rmin (b - x) (x - a))
H:- h < x - a
a < x + h < b
 rewrite Rabs_Ropp; apply Rlt_le_trans with (1 := cmp).f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h0 : R,
h0 <> 0 ->
Rabs h0 < d ->
Rabs ((f (x + h0) - f x) / h0 - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
h:R
hn0:h <> 0
cmp:Rabs h < Rmin d (Rmin (b - x) (x - a))
Rmin d (Rmin (b - x) (x - a)) <= x - a
f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h0 : R,
h0 <> 0 ->
Rabs h0 < d ->
Rabs ((f (x + h0) - f x) / h0 - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
h:R
hn0:h <> 0
cmp:Rabs h < Rmin d (Rmin (b - x) (x - a))
H:- h < x - a
a < x + h < b
 rewrite Rmin_assoc; apply Rmin_r.f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h0 : R,
h0 <> 0 ->
Rabs h0 < d ->
Rabs ((f (x + h0) - f x) / h0 - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
h:R
hn0:h <> 0
cmp:Rabs h < Rmin d (Rmin (b - x) (x - a))
H:- h < x - a
a < x + h < b
assert (h < b - x).f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h0 : R,
h0 <> 0 ->
Rabs h0 < d ->
Rabs ((f (x + h0) - f x) / h0 - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
h:R
hn0:h <> 0
cmp:Rabs h < Rmin d (Rmin (b - x) (x - a))
H:- h < x - a
h < b - x
f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h0 : R,
h0 <> 0 ->
Rabs h0 < d ->
Rabs ((f (x + h0) - f x) / h0 - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
h:R
hn0:h <> 0
cmp:Rabs h < Rmin d (Rmin (b - x) (x - a))
H:- h < x - a
H0:h < b - x
a < x + h < b
 apply Rle_lt_trans with (1 := Rle_abs _).f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h0 : R,
h0 <> 0 ->
Rabs h0 < d ->
Rabs ((f (x + h0) - f x) / h0 - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
h:R
hn0:h <> 0
cmp:Rabs h < Rmin d (Rmin (b - x) (x - a))
H:- h < x - a
Rabs h < b - x
f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h0 : R,
h0 <> 0 ->
Rabs h0 < d ->
Rabs ((f (x + h0) - f x) / h0 - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
h:R
hn0:h <> 0
cmp:Rabs h < Rmin d (Rmin (b - x) (x - a))
H:- h < x - a
H0:h < b - x
a < x + h < b
 apply Rlt_le_trans with (1 := cmp).f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h0 : R,
h0 <> 0 ->
Rabs h0 < d ->
Rabs ((f (x + h0) - f x) / h0 - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
h:R
hn0:h <> 0
cmp:Rabs h < Rmin d (Rmin (b - x) (x - a))
H:- h < x - a
Rmin d (Rmin (b - x) (x - a)) <= b - x
f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h0 : R,
h0 <> 0 ->
Rabs h0 < d ->
Rabs ((f (x + h0) - f x) / h0 - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
h:R
hn0:h <> 0
cmp:Rabs h < Rmin d (Rmin (b - x) (x - a))
H:- h < x - a
H0:h < b - x
a < x + h < b
 rewrite Rmin_comm, <- Rmin_assoc; apply Rmin_l.f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h0 : R,
h0 <> 0 ->
Rabs h0 < d ->
Rabs ((f (x + h0) - f x) / h0 - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
h:R
hn0:h <> 0
cmp:Rabs h < Rmin d (Rmin (b - x) (x - a))
H:- h < x - a
H0:h < b - x
a < x + h < b
split.f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h0 : R,
h0 <> 0 ->
Rabs h0 < d ->
Rabs ((f (x + h0) - f x) / h0 - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
h:R
hn0:h <> 0
cmp:Rabs h < Rmin d (Rmin (b - x) (x - a))
H:- h < x - a
H0:h < b - x
a < x + h
f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h0 : R,
h0 <> 0 ->
Rabs h0 < d ->
Rabs ((f (x + h0) - f x) / h0 - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
h:R
hn0:h <> 0
cmp:Rabs h < Rmin d (Rmin (b - x) (x - a))
H:- h < x - a
H0:h < b - x
x + h < b
 apply (Rplus_lt_reg_l (- h)).f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h0 : R,
h0 <> 0 ->
Rabs h0 < d ->
Rabs ((f (x + h0) - f x) / h0 - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
h:R
hn0:h <> 0
cmp:Rabs h < Rmin d (Rmin (b - x) (x - a))
H:- h < x - a
H0:h < b - x
- h + a < - h + (x + h)
f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h0 : R,
h0 <> 0 ->
Rabs h0 < d ->
Rabs ((f (x + h0) - f x) / h0 - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
h:R
hn0:h <> 0
cmp:Rabs h < Rmin d (Rmin (b - x) (x - a))
H:- h < x - a
H0:h < b - x
x + h < b
 replace ((-h) + (x + h)) with x by ring.f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h0 : R,
h0 <> 0 ->
Rabs h0 < d ->
Rabs ((f (x + h0) - f x) / h0 - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
h:R
hn0:h <> 0
cmp:Rabs h < Rmin d (Rmin (b - x) (x - a))
H:- h < x - a
H0:h < b - x
- h + a < x
f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h0 : R,
h0 <> 0 ->
Rabs h0 < d ->
Rabs ((f (x + h0) - f x) / h0 - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
h:R
hn0:h <> 0
cmp:Rabs h < Rmin d (Rmin (b - x) (x - a))
H:- h < x - a
H0:h < b - x
x + h < b
 apply (Rplus_lt_reg_r (- a)).f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h0 : R,
h0 <> 0 ->
Rabs h0 < d ->
Rabs ((f (x + h0) - f x) / h0 - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
h:R
hn0:h <> 0
cmp:Rabs h < Rmin d (Rmin (b - x) (x - a))
H:- h < x - a
H0:h < b - x
- h + a + - a < x + - a
f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h0 : R,
h0 <> 0 ->
Rabs h0 < d ->
Rabs ((f (x + h0) - f x) / h0 - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
h:R
hn0:h <> 0
cmp:Rabs h < Rmin d (Rmin (b - x) (x - a))
H:- h < x - a
H0:h < b - x
x + h < b
 replace (((-h) + a) + - a) with (-h) by ring.f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h0 : R,
h0 <> 0 ->
Rabs h0 < d ->
Rabs ((f (x + h0) - f x) / h0 - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
h:R
hn0:h <> 0
cmp:Rabs h < Rmin d (Rmin (b - x) (x - a))
H:- h < x - a
H0:h < b - x
- h < x + - a
f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h0 : R,
h0 <> 0 ->
Rabs h0 < d ->
Rabs ((f (x + h0) - f x) / h0 - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
h:R
hn0:h <> 0
cmp:Rabs h < Rmin d (Rmin (b - x) (x - a))
H:- h < x - a
H0:h < b - x
x + h < b
 assumption.f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h0 : R,
h0 <> 0 ->
Rabs h0 < d ->
Rabs ((f (x + h0) - f x) / h0 - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
h:R
hn0:h <> 0
cmp:Rabs h < Rmin d (Rmin (b - x) (x - a))
H:- h < x - a
H0:h < b - x
x + h < b
apply (Rplus_lt_reg_r (- x)).f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h0 : R,
h0 <> 0 ->
Rabs h0 < d ->
Rabs ((f (x + h0) - f x) / h0 - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
h:R
hn0:h <> 0
cmp:Rabs h < Rmin d (Rmin (b - x) (x - a))
H:- h < x - a
H0:h < b - x
x + h + - x < b + - x
replace (x + h + - x) with h by ring.f, g:R -> R
x, a, b, l:R
axb:a < x < b
fg:forall z : R, a < z < b -> f z = g z
df:derivable_pt_lim f x l
e:R
ep:0 < e
d:posreal
pd:forall h0 : R,
h0 <> 0 ->
Rabs h0 < d ->
Rabs ((f (x + h0) - f x) / h0 - l) < e
d'h:0 < Rmin d (Rmin (b - x) (x - a))
h:R
hn0:h <> 0
cmp:Rabs h < Rmin d (Rmin (b - x) (x - a))
H:- h < x - a
H0:h < b - x
h < b + - x
assumption.
Qed.


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

derivability -> continuity

(***********************************)
(**********)
Lemma derivable_derive :
  forall f (x:R) (pr:derivable_pt f x),  exists l : R, derive_pt f x pr = l.forall (f : R -> R) (x : R) (pr : derivable_pt f x),
exists l : R, derive_pt f x pr = l
Proof.forall (f : R -> R) (x : R) (pr : derivable_pt f x),
exists l : R, derive_pt f x pr = l
  intros; exists (proj1_sig pr).f:R -> R
x:R
pr:derivable_pt f x
derive_pt f x pr = proj1_sig pr
  unfold derive_pt; reflexivity.
Qed.

Theorem derivable_continuous_pt :
  forall f (x:R), derivable_pt f x -> continuity_pt f x.forall (f : R -> R) (x : R),
derivable_pt f x -> continuity_pt f x
Proof.forall (f : R -> R) (x : R),
derivable_pt f x -> continuity_pt f x
  intros f x X.f:R -> R
x:R
X:derivable_pt f x
continuity_pt f x
  generalize (derivable_derive f x X); intro.f:R -> R
x:R
X:derivable_pt f x
H:exists l : R, derive_pt f x X = l
continuity_pt f x
  elim H; intros l H1.f:R -> R
x:R
X:derivable_pt f x
H:exists l0 : R, derive_pt f x X = l0
l:R
H1:derive_pt f x X = l
continuity_pt f x
  cut (l = fct_cte l x).f:R -> R
x:R
X:derivable_pt f x
H:exists l0 : R, derive_pt f x X = l0
l:R
H1:derive_pt f x X = l
l = fct_cte l x -> continuity_pt f x
f:R -> R
x:R
X:derivable_pt f x
H:exists l0 : R, derive_pt f x X = l0
l:R
H1:derive_pt f x X = l
l = fct_cte l x
  intro.f:R -> R
x:R
X:derivable_pt f x
H:exists l0 : R, derive_pt f x X = l0
l:R
H1:derive_pt f x X = l
H0:l = fct_cte l x
continuity_pt f x
f:R -> R
x:R
X:derivable_pt f x
H:exists l0 : R, derive_pt f x X = l0
l:R
H1:derive_pt f x X = l
l = fct_cte l x
  rewrite H0 in H1.f:R -> R
x:R
X:derivable_pt f x
H:exists l0 : R, derive_pt f x X = l0
l:R
H1:derive_pt f x X = fct_cte l x
H0:l = fct_cte l x
continuity_pt f x
f:R -> R
x:R
X:derivable_pt f x
H:exists l0 : R, derive_pt f x X = l0
l:R
H1:derive_pt f x X = l
l = fct_cte l x
  generalize (derive_pt_D_in f (fct_cte l) x); intro.f:R -> R
x:R
X:derivable_pt f x
H:exists l0 : R, derive_pt f x X = l0
l:R
H1:derive_pt f x X = fct_cte l x
H0:l = fct_cte l x
H2:forall pr : derivable_pt f x,
D_in f (fct_cte l) no_cond x <->
derive_pt f x pr = fct_cte l x
continuity_pt f x
f:R -> R
x:R
X:derivable_pt f x
H:exists l0 : R, derive_pt f x X = l0
l:R
H1:derive_pt f x X = l
l = fct_cte l x
  elim (H2 X); intros.f:R -> R
x:R
X:derivable_pt f x
H:exists l0 : R, derive_pt f x X = l0
l:R
H1:derive_pt f x X = fct_cte l x
H0:l = fct_cte l x
H2:forall pr : derivable_pt f x,
D_in f (fct_cte l) no_cond x <->
derive_pt f x pr = fct_cte l x
H3:D_in f (fct_cte l) no_cond x ->
derive_pt f x X = fct_cte l x
H4:derive_pt f x X = fct_cte l x ->
D_in f (fct_cte l) no_cond x
continuity_pt f x
f:R -> R
x:R
X:derivable_pt f x
H:exists l0 : R, derive_pt f x X = l0
l:R
H1:derive_pt f x X = l
l = fct_cte l x
  generalize (H4 H1); intro.f:R -> R
x:R
X:derivable_pt f x
H:exists l0 : R, derive_pt f x X = l0
l:R
H1:derive_pt f x X = fct_cte l x
H0:l = fct_cte l x
H2:forall pr : derivable_pt f x,
D_in f (fct_cte l) no_cond x <->
derive_pt f x pr = fct_cte l x
H3:D_in f (fct_cte l) no_cond x ->
derive_pt f x X = fct_cte l x
H4:derive_pt f x X = fct_cte l x ->
D_in f (fct_cte l) no_cond x
H5:D_in f (fct_cte l) no_cond x
continuity_pt f x
f:R -> R
x:R
X:derivable_pt f x
H:exists l0 : R, derive_pt f x X = l0
l:R
H1:derive_pt f x X = l
l = fct_cte l x
  unfold continuity_pt.f:R -> R
x:R
X:derivable_pt f x
H:exists l0 : R, derive_pt f x X = l0
l:R
H1:derive_pt f x X = fct_cte l x
H0:l = fct_cte l x
H2:forall pr : derivable_pt f x,
D_in f (fct_cte l) no_cond x <->
derive_pt f x pr = fct_cte l x
H3:D_in f (fct_cte l) no_cond x ->
derive_pt f x X = fct_cte l x
H4:derive_pt f x X = fct_cte l x ->
D_in f (fct_cte l) no_cond x
H5:D_in f (fct_cte l) no_cond x
continue_in f no_cond x
f:R -> R
x:R
X:derivable_pt f x
H:exists l0 : R, derive_pt f x X = l0
l:R
H1:derive_pt f x X = l
l = fct_cte l x
  apply (cont_deriv f (fct_cte l) no_cond x H5).f:R -> R
x:R
X:derivable_pt f x
H:exists l0 : R, derive_pt f x X = l0
l:R
H1:derive_pt f x X = l
l = fct_cte l x
  unfold fct_cte; reflexivity.
Qed.

Theorem derivable_continuous : forall f, derivable f -> continuity f.forall f : R -> R, derivable f -> continuity f
Proof.forall f : R -> R, derivable f -> continuity f
  unfold derivable, continuity; intros f X x.f:R -> R
X:forall x0 : R, derivable_pt f x0
x:R
continuity_pt f x
  apply (derivable_continuous_pt f x (X x)).
Qed.

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

Main rules

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

Lemma derivable_pt_lim_plus :
  forall f1 f2 (x l1 l2:R),
    derivable_pt_lim f1 x l1 ->
    derivable_pt_lim f2 x l2 -> derivable_pt_lim (f1 + f2) x (l1 + l2).forall (f1 f2 : R -> R) (x l1 l2 : R),
derivable_pt_lim f1 x l1 ->
derivable_pt_lim f2 x l2 ->
derivable_pt_lim (f1 + f2) x (l1 + l2)
  intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
derivable_pt_lim (f1 + f2) x (l1 + l2)
  apply uniqueness_step3.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
limit1_in
  (fun h : R =>
   ((f1 + f2)%F (x + h) - (f1 + f2)%F x) / h)
  (fun h : R => h <> 0) (l1 + l2) 0
  assert (H1 := uniqueness_step2 _ _ _ H).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
limit1_in
  (fun h : R =>
   ((f1 + f2)%F (x + h) - (f1 + f2)%F x) / h)
  (fun h : R => h <> 0) (l1 + l2) 0
  assert (H2 := uniqueness_step2 _ _ _ H0).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
limit1_in
  (fun h : R =>
   ((f1 + f2)%F (x + h) - (f1 + f2)%F x) / h)
  (fun h : R => h <> 0) (l1 + l2) 0
  unfold plus_fct.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
limit1_in
  (fun h : R =>
   (f1 (x + h) + f2 (x + h) - (f1 x + f2 x)) / h)
  (fun h : R => h <> 0) (l1 + l2) 0
  cut
    (forall h:R,
      (f1 (x + h) + f2 (x + h) - (f1 x + f2 x)) / h =
      (f1 (x + h) - f1 x) / h + (f2 (x + h) - f2 x) / h).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
(forall h : R,
 (f1 (x + h) + f2 (x + h) - (f1 x + f2 x)) / h =
 (f1 (x + h) - f1 x) / h + (f2 (x + h) - f2 x) / h) ->
limit1_in
  (fun h : R =>
   (f1 (x + h) + f2 (x + h) - (f1 x + f2 x)) / h)
  (fun h : R => h <> 0) (l1 + l2) 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
forall h : R,
(f1 (x + h) + f2 (x + h) - (f1 x + f2 x)) / h =
(f1 (x + h) - f1 x) / h + (f2 (x + h) - f2 x) / h
  intro.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
H3:forall h : R,
(f1 (x + h) + f2 (x + h) - (f1 x + f2 x)) / h =
(f1 (x + h) - f1 x) / h + (f2 (x + h) - f2 x) / h
limit1_in
  (fun h : R =>
   (f1 (x + h) + f2 (x + h) - (f1 x + f2 x)) / h)
  (fun h : R => h <> 0) (l1 + l2) 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
forall h : R,
(f1 (x + h) + f2 (x + h) - (f1 x + f2 x)) / h =
(f1 (x + h) - f1 x) / h + (f2 (x + h) - f2 x) / h
  generalize
    (limit_plus (fun h':R => (f1 (x + h') - f1 x) / h')
      (fun h':R => (f2 (x + h') - f2 x) / h') (fun h:R => h <> 0) l1 l2 0 H1 H2).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
H3:forall h : R,
(f1 (x + h) + f2 (x + h) - (f1 x + f2 x)) / h =
(f1 (x + h) - f1 x) / h + (f2 (x + h) - f2 x) / h
limit1_in
  (fun x0 : R =>
   (f1 (x + x0) - f1 x) / x0 +
   (f2 (x + x0) - f2 x) / x0) (fun h : R => h <> 0)
  (l1 + l2) 0 ->
limit1_in
  (fun h : R =>
   (f1 (x + h) + f2 (x + h) - (f1 x + f2 x)) / h)
  (fun h : R => h <> 0) (l1 + l2) 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
forall h : R,
(f1 (x + h) + f2 (x + h) - (f1 x + f2 x)) / h =
(f1 (x + h) - f1 x) / h + (f2 (x + h) - f2 x) / h
  unfold limit1_in; unfold limit_in; unfold dist;
    simpl; unfold R_dist; intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
H3:forall h : R,
(f1 (x + h) + f2 (x + h) - (f1 x + f2 x)) / h =
(f1 (x + h) - f1 x) / h + (f2 (x + h) - f2 x) / h
H4:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   x0 <> 0 /\ Rabs (x0 - 0) < alp ->
   Rabs
     ((f1 (x + x0) - f1 x) / x0 +
      (f2 (x + x0) - f2 x) / x0 - (l1 + l2)) <
   eps0)
eps:R
H5:eps > 0
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   x0 <> 0 /\ Rabs (x0 - 0) < alp ->
   Rabs
     ((f1 (x + x0) + f2 (x + x0) - (f1 x + f2 x)) / x0 -
      (l1 + l2)) < eps)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
forall h : R,
(f1 (x + h) + f2 (x + h) - (f1 x + f2 x)) / h =
(f1 (x + h) - f1 x) / h + (f2 (x + h) - f2 x) / h
  elim (H4 eps H5); intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
H3:forall h : R,
(f1 (x + h) + f2 (x + h) - (f1 x + f2 x)) / h =
(f1 (x + h) - f1 x) / h + (f2 (x + h) - f2 x) / h
H4:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   x1 <> 0 /\ Rabs (x1 - 0) < alp ->
   Rabs
     ((f1 (x + x1) - f1 x) / x1 +
      (f2 (x + x1) - f2 x) / x1 - (l1 + l2)) <
   eps0)
eps:R
H5:eps > 0
x0:R
H6:x0 > 0 /\
(forall x1 : R,
 x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
 Rabs
   ((f1 (x + x1) - f1 x) / x1 +
    (f2 (x + x1) - f2 x) / x1 - (l1 + l2)) < eps)
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   x1 <> 0 /\ Rabs (x1 - 0) < alp ->
   Rabs
     ((f1 (x + x1) + f2 (x + x1) - (f1 x + f2 x)) / x1 -
      (l1 + l2)) < eps)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
forall h : R,
(f1 (x + h) + f2 (x + h) - (f1 x + f2 x)) / h =
(f1 (x + h) - f1 x) / h + (f2 (x + h) - f2 x) / h
  exists x0.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
H3:forall h : R,
(f1 (x + h) + f2 (x + h) - (f1 x + f2 x)) / h =
(f1 (x + h) - f1 x) / h + (f2 (x + h) - f2 x) / h
H4:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   x1 <> 0 /\ Rabs (x1 - 0) < alp ->
   Rabs
     ((f1 (x + x1) - f1 x) / x1 +
      (f2 (x + x1) - f2 x) / x1 - (l1 + l2)) <
   eps0)
eps:R
H5:eps > 0
x0:R
H6:x0 > 0 /\
(forall x1 : R,
 x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
 Rabs
   ((f1 (x + x1) - f1 x) / x1 +
    (f2 (x + x1) - f2 x) / x1 - (l1 + l2)) < eps)
x0 > 0 /\
(forall x1 : R,
 x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
 Rabs
   ((f1 (x + x1) + f2 (x + x1) - (f1 x + f2 x)) / x1 -
    (l1 + l2)) < eps)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
forall h : R,
(f1 (x + h) + f2 (x + h) - (f1 x + f2 x)) / h =
(f1 (x + h) - f1 x) / h + (f2 (x + h) - f2 x) / h
  elim H6; intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
H3:forall h : R,
(f1 (x + h) + f2 (x + h) - (f1 x + f2 x)) / h =
(f1 (x + h) - f1 x) / h + (f2 (x + h) - f2 x) / h
H4:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   x1 <> 0 /\ Rabs (x1 - 0) < alp ->
   Rabs
     ((f1 (x + x1) - f1 x) / x1 +
      (f2 (x + x1) - f2 x) / x1 - (l1 + l2)) <
   eps0)
eps:R
H5:eps > 0
x0:R
H6:x0 > 0 /\
(forall x1 : R,
 x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
 Rabs
   ((f1 (x + x1) - f1 x) / x1 +
    (f2 (x + x1) - f2 x) / x1 - (l1 + l2)) < eps)
H7:x0 > 0
H8:forall x1 : R,
x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
Rabs
  ((f1 (x + x1) - f1 x) / x1 +
   (f2 (x + x1) - f2 x) / x1 - (l1 + l2)) < eps
x0 > 0 /\
(forall x1 : R,
 x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
 Rabs
   ((f1 (x + x1) + f2 (x + x1) - (f1 x + f2 x)) / x1 -
    (l1 + l2)) < eps)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
forall h : R,
(f1 (x + h) + f2 (x + h) - (f1 x + f2 x)) / h =
(f1 (x + h) - f1 x) / h + (f2 (x + h) - f2 x) / h
  split.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
H3:forall h : R,
(f1 (x + h) + f2 (x + h) - (f1 x + f2 x)) / h =
(f1 (x + h) - f1 x) / h + (f2 (x + h) - f2 x) / h
H4:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   x1 <> 0 /\ Rabs (x1 - 0) < alp ->
   Rabs
     ((f1 (x + x1) - f1 x) / x1 +
      (f2 (x + x1) - f2 x) / x1 - (l1 + l2)) <
   eps0)
eps:R
H5:eps > 0
x0:R
H6:x0 > 0 /\
(forall x1 : R,
 x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
 Rabs
   ((f1 (x + x1) - f1 x) / x1 +
    (f2 (x + x1) - f2 x) / x1 - (l1 + l2)) < eps)
H7:x0 > 0
H8:forall x1 : R,
x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
Rabs
  ((f1 (x + x1) - f1 x) / x1 +
   (f2 (x + x1) - f2 x) / x1 - (l1 + l2)) < eps
x0 > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
H3:forall h : R,
(f1 (x + h) + f2 (x + h) - (f1 x + f2 x)) / h =
(f1 (x + h) - f1 x) / h + (f2 (x + h) - f2 x) / h
H4:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   x1 <> 0 /\ Rabs (x1 - 0) < alp ->
   Rabs
     ((f1 (x + x1) - f1 x) / x1 +
      (f2 (x + x1) - f2 x) / x1 - (l1 + l2)) <
   eps0)
eps:R
H5:eps > 0
x0:R
H6:x0 > 0 /\
(forall x1 : R,
 x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
 Rabs
   ((f1 (x + x1) - f1 x) / x1 +
    (f2 (x + x1) - f2 x) / x1 - (l1 + l2)) < eps)
H7:x0 > 0
H8:forall x1 : R,
x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
Rabs
  ((f1 (x + x1) - f1 x) / x1 +
   (f2 (x + x1) - f2 x) / x1 - (l1 + l2)) < eps
forall x1 : R,
x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
Rabs
  ((f1 (x + x1) + f2 (x + x1) - (f1 x + f2 x)) / x1 -
   (l1 + l2)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
forall h : R,
(f1 (x + h) + f2 (x + h) - (f1 x + f2 x)) / h =
(f1 (x + h) - f1 x) / h + (f2 (x + h) - f2 x) / h
  assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
H3:forall h : R,
(f1 (x + h) + f2 (x + h) - (f1 x + f2 x)) / h =
(f1 (x + h) - f1 x) / h + (f2 (x + h) - f2 x) / h
H4:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   x1 <> 0 /\ Rabs (x1 - 0) < alp ->
   Rabs
     ((f1 (x + x1) - f1 x) / x1 +
      (f2 (x + x1) - f2 x) / x1 - (l1 + l2)) <
   eps0)
eps:R
H5:eps > 0
x0:R
H6:x0 > 0 /\
(forall x1 : R,
 x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
 Rabs
   ((f1 (x + x1) - f1 x) / x1 +
    (f2 (x + x1) - f2 x) / x1 - (l1 + l2)) < eps)
H7:x0 > 0
H8:forall x1 : R,
x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
Rabs
  ((f1 (x + x1) - f1 x) / x1 +
   (f2 (x + x1) - f2 x) / x1 - (l1 + l2)) < eps
forall x1 : R,
x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
Rabs
  ((f1 (x + x1) + f2 (x + x1) - (f1 x + f2 x)) / x1 -
   (l1 + l2)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
forall h : R,
(f1 (x + h) + f2 (x + h) - (f1 x + f2 x)) / h =
(f1 (x + h) - f1 x) / h + (f2 (x + h) - f2 x) / h
  intros; rewrite H3; apply H8; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
forall h : R,
(f1 (x + h) + f2 (x + h) - (f1 x + f2 x)) / h =
(f1 (x + h) - f1 x) / h + (f2 (x + h) - f2 x) / h
  intro; unfold Rdiv; ring.
Qed.

Lemma derivable_pt_lim_opp :
  forall f (x l:R), derivable_pt_lim f x l -> derivable_pt_lim (- f) x (- l).forall (f : R -> R) (x l : R),
derivable_pt_lim f x l ->
derivable_pt_lim (- f) x (- l)
Proof.forall (f : R -> R) (x l : R),
derivable_pt_lim f x l ->
derivable_pt_lim (- f) x (- l)
  intros.f:R -> R
x, l:R
H:derivable_pt_lim f x l
derivable_pt_lim (- f) x (- l)
  apply uniqueness_step3.f:R -> R
x, l:R
H:derivable_pt_lim f x l
limit1_in
  (fun h : R => ((- f)%F (x + h) - (- f)%F x) / h)
  (fun h : R => h <> 0) (- l) 0
  assert (H1 := uniqueness_step2 _ _ _ H).f:R -> R
x, l:R
H:derivable_pt_lim f x l
H1:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l 0
limit1_in
  (fun h : R => ((- f)%F (x + h) - (- f)%F x) / h)
  (fun h : R => h <> 0) (- l) 0
  unfold opp_fct.f:R -> R
x, l:R
H:derivable_pt_lim f x l
H1:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l 0
limit1_in (fun h : R => (- f (x + h) - - f x) / h)
  (fun h : R => h <> 0) (- l) 0
  cut (forall h:R, (- f (x + h) - - f x) / h = - ((f (x + h) - f x) / h)).f:R -> R
x, l:R
H:derivable_pt_lim f x l
H1:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l 0
(forall h : R,
 (- f (x + h) - - f x) / h = - ((f (x + h) - f x) / h)) ->
limit1_in (fun h : R => (- f (x + h) - - f x) / h)
  (fun h : R => h <> 0) (- l) 0
f:R -> R
x, l:R
H:derivable_pt_lim f x l
H1:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l 0
forall h : R,
(- f (x + h) - - f x) / h = - ((f (x + h) - f x) / h)
  intro.f:R -> R
x, l:R
H:derivable_pt_lim f x l
H1:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l 0
H0:forall h : R,
(- f (x + h) - - f x) / h =
- ((f (x + h) - f x) / h)
limit1_in (fun h : R => (- f (x + h) - - f x) / h)
  (fun h : R => h <> 0) (- l) 0
f:R -> R
x, l:R
H:derivable_pt_lim f x l
H1:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l 0
forall h : R,
(- f (x + h) - - f x) / h = - ((f (x + h) - f x) / h)
  generalize
    (limit_Ropp (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) l 0 H1).f:R -> R
x, l:R
H:derivable_pt_lim f x l
H1:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l 0
H0:forall h : R,
(- f (x + h) - - f x) / h =
- ((f (x + h) - f x) / h)
limit1_in (fun x0 : R => - ((f (x + x0) - f x) / x0))
  (fun h : R => h <> 0) (- l) 0 ->
limit1_in (fun h : R => (- f (x + h) - - f x) / h)
  (fun h : R => h <> 0) (- l) 0
f:R -> R
x, l:R
H:derivable_pt_lim f x l
H1:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l 0
forall h : R,
(- f (x + h) - - f x) / h = - ((f (x + h) - f x) / h)
  unfold limit1_in; unfold limit_in; unfold dist;
    simpl; unfold R_dist; intros.f:R -> R
x, l:R
H:derivable_pt_lim f x l
H1:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l 0
H0:forall h : R,
(- f (x + h) - - f x) / h =
- ((f (x + h) - f x) / h)
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   x0 <> 0 /\ Rabs (x0 - 0) < alp ->
   Rabs (- ((f (x + x0) - f x) / x0) - - l) <
   eps0)
eps:R
H3:eps > 0
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   x0 <> 0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((- f (x + x0) - - f x) / x0 - - l) < eps)
f:R -> R
x, l:R
H:derivable_pt_lim f x l
H1:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l 0
forall h : R,
(- f (x + h) - - f x) / h = - ((f (x + h) - f x) / h)
  elim (H2 eps H3); intros.f:R -> R
x, l:R
H:derivable_pt_lim f x l
H1:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l 0
H0:forall h : R,
(- f (x + h) - - f x) / h =
- ((f (x + h) - f x) / h)
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   x1 <> 0 /\ Rabs (x1 - 0) < alp ->
   Rabs (- ((f (x + x1) - f x) / x1) - - l) <
   eps0)
eps:R
H3:eps > 0
x0:R
H4:x0 > 0 /\
(forall x1 : R,
 x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
 Rabs (- ((f (x + x1) - f x) / x1) - - l) < eps)
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   x1 <> 0 /\ Rabs (x1 - 0) < alp ->
   Rabs ((- f (x + x1) - - f x) / x1 - - l) < eps)
f:R -> R
x, l:R
H:derivable_pt_lim f x l
H1:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l 0
forall h : R,
(- f (x + h) - - f x) / h = - ((f (x + h) - f x) / h)
  exists x0.f:R -> R
x, l:R
H:derivable_pt_lim f x l
H1:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l 0
H0:forall h : R,
(- f (x + h) - - f x) / h =
- ((f (x + h) - f x) / h)
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   x1 <> 0 /\ Rabs (x1 - 0) < alp ->
   Rabs (- ((f (x + x1) - f x) / x1) - - l) <
   eps0)
eps:R
H3:eps > 0
x0:R
H4:x0 > 0 /\
(forall x1 : R,
 x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
 Rabs (- ((f (x + x1) - f x) / x1) - - l) < eps)
x0 > 0 /\
(forall x1 : R,
 x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
 Rabs ((- f (x + x1) - - f x) / x1 - - l) < eps)
f:R -> R
x, l:R
H:derivable_pt_lim f x l
H1:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l 0
forall h : R,
(- f (x + h) - - f x) / h = - ((f (x + h) - f x) / h)
  elim H4; intros.f:R -> R
x, l:R
H:derivable_pt_lim f x l
H1:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l 0
H0:forall h : R,
(- f (x + h) - - f x) / h =
- ((f (x + h) - f x) / h)
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   x1 <> 0 /\ Rabs (x1 - 0) < alp ->
   Rabs (- ((f (x + x1) - f x) / x1) - - l) <
   eps0)
eps:R
H3:eps > 0
x0:R
H4:x0 > 0 /\
(forall x1 : R,
 x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
 Rabs (- ((f (x + x1) - f x) / x1) - - l) < eps)
H5:x0 > 0
H6:forall x1 : R,
x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
Rabs (- ((f (x + x1) - f x) / x1) - - l) < eps
x0 > 0 /\
(forall x1 : R,
 x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
 Rabs ((- f (x + x1) - - f x) / x1 - - l) < eps)
f:R -> R
x, l:R
H:derivable_pt_lim f x l
H1:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l 0
forall h : R,
(- f (x + h) - - f x) / h = - ((f (x + h) - f x) / h)
  split.f:R -> R
x, l:R
H:derivable_pt_lim f x l
H1:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l 0
H0:forall h : R,
(- f (x + h) - - f x) / h =
- ((f (x + h) - f x) / h)
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   x1 <> 0 /\ Rabs (x1 - 0) < alp ->
   Rabs (- ((f (x + x1) - f x) / x1) - - l) <
   eps0)
eps:R
H3:eps > 0
x0:R
H4:x0 > 0 /\
(forall x1 : R,
 x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
 Rabs (- ((f (x + x1) - f x) / x1) - - l) < eps)
H5:x0 > 0
H6:forall x1 : R,
x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
Rabs (- ((f (x + x1) - f x) / x1) - - l) < eps
x0 > 0
f:R -> R
x, l:R
H:derivable_pt_lim f x l
H1:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l 0
H0:forall h : R,
(- f (x + h) - - f x) / h =
- ((f (x + h) - f x) / h)
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   x1 <> 0 /\ Rabs (x1 - 0) < alp ->
   Rabs (- ((f (x + x1) - f x) / x1) - - l) <
   eps0)
eps:R
H3:eps > 0
x0:R
H4:x0 > 0 /\
(forall x1 : R,
 x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
 Rabs (- ((f (x + x1) - f x) / x1) - - l) < eps)
H5:x0 > 0
H6:forall x1 : R,
x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
Rabs (- ((f (x + x1) - f x) / x1) - - l) < eps
forall x1 : R,
x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
Rabs ((- f (x + x1) - - f x) / x1 - - l) < eps
f:R -> R
x, l:R
H:derivable_pt_lim f x l
H1:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l 0
forall h : R,
(- f (x + h) - - f x) / h = - ((f (x + h) - f x) / h)
  assumption.f:R -> R
x, l:R
H:derivable_pt_lim f x l
H1:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l 0
H0:forall h : R,
(- f (x + h) - - f x) / h =
- ((f (x + h) - f x) / h)
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   x1 <> 0 /\ Rabs (x1 - 0) < alp ->
   Rabs (- ((f (x + x1) - f x) / x1) - - l) <
   eps0)
eps:R
H3:eps > 0
x0:R
H4:x0 > 0 /\
(forall x1 : R,
 x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
 Rabs (- ((f (x + x1) - f x) / x1) - - l) < eps)
H5:x0 > 0
H6:forall x1 : R,
x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
Rabs (- ((f (x + x1) - f x) / x1) - - l) < eps
forall x1 : R,
x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
Rabs ((- f (x + x1) - - f x) / x1 - - l) < eps
f:R -> R
x, l:R
H:derivable_pt_lim f x l
H1:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l 0
forall h : R,
(- f (x + h) - - f x) / h = - ((f (x + h) - f x) / h)
  intros; rewrite H0; apply H6; assumption.f:R -> R
x, l:R
H:derivable_pt_lim f x l
H1:limit1_in (fun h : R => (f (x + h) - f x) / h)
  (fun h : R => h <> 0) l 0
forall h : R,
(- f (x + h) - - f x) / h = - ((f (x + h) - f x) / h)
  intro; unfold Rdiv; ring.
Qed.

Lemma derivable_pt_lim_minus :
  forall f1 f2 (x l1 l2:R),
    derivable_pt_lim f1 x l1 ->
    derivable_pt_lim f2 x l2 -> derivable_pt_lim (f1 - f2) x (l1 - l2).forall (f1 f2 : R -> R) (x l1 l2 : R),
derivable_pt_lim f1 x l1 ->
derivable_pt_lim f2 x l2 ->
derivable_pt_lim (f1 - f2) x (l1 - l2)
Proof.forall (f1 f2 : R -> R) (x l1 l2 : R),
derivable_pt_lim f1 x l1 ->
derivable_pt_lim f2 x l2 ->
derivable_pt_lim (f1 - f2) x (l1 - l2)
  intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
derivable_pt_lim (f1 - f2) x (l1 - l2)
  apply uniqueness_step3.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
limit1_in
  (fun h : R =>
   ((f1 - f2)%F (x + h) - (f1 - f2)%F x) / h)
  (fun h : R => h <> 0) (l1 - l2) 0
  assert (H1 := uniqueness_step2 _ _ _ H).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
limit1_in
  (fun h : R =>
   ((f1 - f2)%F (x + h) - (f1 - f2)%F x) / h)
  (fun h : R => h <> 0) (l1 - l2) 0
  assert (H2 := uniqueness_step2 _ _ _ H0).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
limit1_in
  (fun h : R =>
   ((f1 - f2)%F (x + h) - (f1 - f2)%F x) / h)
  (fun h : R => h <> 0) (l1 - l2) 0
  unfold minus_fct.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
limit1_in
  (fun h : R =>
   (f1 (x + h) - f2 (x + h) - (f1 x - f2 x)) / h)
  (fun h : R => h <> 0) (l1 - l2) 0
  cut
    (forall h:R,
      (f1 (x + h) - f1 x) / h - (f2 (x + h) - f2 x) / h =
      (f1 (x + h) - f2 (x + h) - (f1 x - f2 x)) / h).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
(forall h : R,
 (f1 (x + h) - f1 x) / h - (f2 (x + h) - f2 x) / h =
 (f1 (x + h) - f2 (x + h) - (f1 x - f2 x)) / h) ->
limit1_in
  (fun h : R =>
   (f1 (x + h) - f2 (x + h) - (f1 x - f2 x)) / h)
  (fun h : R => h <> 0) (l1 - l2) 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
forall h : R,
(f1 (x + h) - f1 x) / h - (f2 (x + h) - f2 x) / h =
(f1 (x + h) - f2 (x + h) - (f1 x - f2 x)) / h
  intro.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
H3:forall h : R,
(f1 (x + h) - f1 x) / h - (f2 (x + h) - f2 x) / h =
(f1 (x + h) - f2 (x + h) - (f1 x - f2 x)) / h
limit1_in
  (fun h : R =>
   (f1 (x + h) - f2 (x + h) - (f1 x - f2 x)) / h)
  (fun h : R => h <> 0) (l1 - l2) 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
forall h : R,
(f1 (x + h) - f1 x) / h - (f2 (x + h) - f2 x) / h =
(f1 (x + h) - f2 (x + h) - (f1 x - f2 x)) / h
  generalize
    (limit_minus (fun h':R => (f1 (x + h') - f1 x) / h')
      (fun h':R => (f2 (x + h') - f2 x) / h') (fun h:R => h <> 0) l1 l2 0 H1 H2).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
H3:forall h : R,
(f1 (x + h) - f1 x) / h - (f2 (x + h) - f2 x) / h =
(f1 (x + h) - f2 (x + h) - (f1 x - f2 x)) / h
limit1_in
  (fun x0 : R =>
   (f1 (x + x0) - f1 x) / x0 -
   (f2 (x + x0) - f2 x) / x0) (fun h : R => h <> 0)
  (l1 - l2) 0 ->
limit1_in
  (fun h : R =>
   (f1 (x + h) - f2 (x + h) - (f1 x - f2 x)) / h)
  (fun h : R => h <> 0) (l1 - l2) 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
forall h : R,
(f1 (x + h) - f1 x) / h - (f2 (x + h) - f2 x) / h =
(f1 (x + h) - f2 (x + h) - (f1 x - f2 x)) / h
  unfold limit1_in; unfold limit_in; unfold dist;
    simpl; unfold R_dist; intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
H3:forall h : R,
(f1 (x + h) - f1 x) / h - (f2 (x + h) - f2 x) / h =
(f1 (x + h) - f2 (x + h) - (f1 x - f2 x)) / h
H4:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   x0 <> 0 /\ Rabs (x0 - 0) < alp ->
   Rabs
     ((f1 (x + x0) - f1 x) / x0 -
      (f2 (x + x0) - f2 x) / x0 - (l1 - l2)) <
   eps0)
eps:R
H5:eps > 0
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   x0 <> 0 /\ Rabs (x0 - 0) < alp ->
   Rabs
     ((f1 (x + x0) - f2 (x + x0) - (f1 x - f2 x)) / x0 -
      (l1 - l2)) < eps)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
forall h : R,
(f1 (x + h) - f1 x) / h - (f2 (x + h) - f2 x) / h =
(f1 (x + h) - f2 (x + h) - (f1 x - f2 x)) / h
  elim (H4 eps H5); intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
H3:forall h : R,
(f1 (x + h) - f1 x) / h - (f2 (x + h) - f2 x) / h =
(f1 (x + h) - f2 (x + h) - (f1 x - f2 x)) / h
H4:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   x1 <> 0 /\ Rabs (x1 - 0) < alp ->
   Rabs
     ((f1 (x + x1) - f1 x) / x1 -
      (f2 (x + x1) - f2 x) / x1 - (l1 - l2)) <
   eps0)
eps:R
H5:eps > 0
x0:R
H6:x0 > 0 /\
(forall x1 : R,
 x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
 Rabs
   ((f1 (x + x1) - f1 x) / x1 -
    (f2 (x + x1) - f2 x) / x1 - (l1 - l2)) < eps)
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   x1 <> 0 /\ Rabs (x1 - 0) < alp ->
   Rabs
     ((f1 (x + x1) - f2 (x + x1) - (f1 x - f2 x)) / x1 -
      (l1 - l2)) < eps)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
forall h : R,
(f1 (x + h) - f1 x) / h - (f2 (x + h) - f2 x) / h =
(f1 (x + h) - f2 (x + h) - (f1 x - f2 x)) / h
  exists x0.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
H3:forall h : R,
(f1 (x + h) - f1 x) / h - (f2 (x + h) - f2 x) / h =
(f1 (x + h) - f2 (x + h) - (f1 x - f2 x)) / h
H4:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   x1 <> 0 /\ Rabs (x1 - 0) < alp ->
   Rabs
     ((f1 (x + x1) - f1 x) / x1 -
      (f2 (x + x1) - f2 x) / x1 - (l1 - l2)) <
   eps0)
eps:R
H5:eps > 0
x0:R
H6:x0 > 0 /\
(forall x1 : R,
 x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
 Rabs
   ((f1 (x + x1) - f1 x) / x1 -
    (f2 (x + x1) - f2 x) / x1 - (l1 - l2)) < eps)
x0 > 0 /\
(forall x1 : R,
 x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
 Rabs
   ((f1 (x + x1) - f2 (x + x1) - (f1 x - f2 x)) / x1 -
    (l1 - l2)) < eps)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
forall h : R,
(f1 (x + h) - f1 x) / h - (f2 (x + h) - f2 x) / h =
(f1 (x + h) - f2 (x + h) - (f1 x - f2 x)) / h
  elim H6; intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
H3:forall h : R,
(f1 (x + h) - f1 x) / h - (f2 (x + h) - f2 x) / h =
(f1 (x + h) - f2 (x + h) - (f1 x - f2 x)) / h
H4:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   x1 <> 0 /\ Rabs (x1 - 0) < alp ->
   Rabs
     ((f1 (x + x1) - f1 x) / x1 -
      (f2 (x + x1) - f2 x) / x1 - (l1 - l2)) <
   eps0)
eps:R
H5:eps > 0
x0:R
H6:x0 > 0 /\
(forall x1 : R,
 x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
 Rabs
   ((f1 (x + x1) - f1 x) / x1 -
    (f2 (x + x1) - f2 x) / x1 - (l1 - l2)) < eps)
H7:x0 > 0
H8:forall x1 : R,
x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
Rabs
  ((f1 (x + x1) - f1 x) / x1 -
   (f2 (x + x1) - f2 x) / x1 - (l1 - l2)) < eps
x0 > 0 /\
(forall x1 : R,
 x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
 Rabs
   ((f1 (x + x1) - f2 (x + x1) - (f1 x - f2 x)) / x1 -
    (l1 - l2)) < eps)
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
forall h : R,
(f1 (x + h) - f1 x) / h - (f2 (x + h) - f2 x) / h =
(f1 (x + h) - f2 (x + h) - (f1 x - f2 x)) / h
  split.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
H3:forall h : R,
(f1 (x + h) - f1 x) / h - (f2 (x + h) - f2 x) / h =
(f1 (x + h) - f2 (x + h) - (f1 x - f2 x)) / h
H4:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   x1 <> 0 /\ Rabs (x1 - 0) < alp ->
   Rabs
     ((f1 (x + x1) - f1 x) / x1 -
      (f2 (x + x1) - f2 x) / x1 - (l1 - l2)) <
   eps0)
eps:R
H5:eps > 0
x0:R
H6:x0 > 0 /\
(forall x1 : R,
 x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
 Rabs
   ((f1 (x + x1) - f1 x) / x1 -
    (f2 (x + x1) - f2 x) / x1 - (l1 - l2)) < eps)
H7:x0 > 0
H8:forall x1 : R,
x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
Rabs
  ((f1 (x + x1) - f1 x) / x1 -
   (f2 (x + x1) - f2 x) / x1 - (l1 - l2)) < eps
x0 > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
H3:forall h : R,
(f1 (x + h) - f1 x) / h - (f2 (x + h) - f2 x) / h =
(f1 (x + h) - f2 (x + h) - (f1 x - f2 x)) / h
H4:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   x1 <> 0 /\ Rabs (x1 - 0) < alp ->
   Rabs
     ((f1 (x + x1) - f1 x) / x1 -
      (f2 (x + x1) - f2 x) / x1 - (l1 - l2)) <
   eps0)
eps:R
H5:eps > 0
x0:R
H6:x0 > 0 /\
(forall x1 : R,
 x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
 Rabs
   ((f1 (x + x1) - f1 x) / x1 -
    (f2 (x + x1) - f2 x) / x1 - (l1 - l2)) < eps)
H7:x0 > 0
H8:forall x1 : R,
x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
Rabs
  ((f1 (x + x1) - f1 x) / x1 -
   (f2 (x + x1) - f2 x) / x1 - (l1 - l2)) < eps
forall x1 : R,
x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
Rabs
  ((f1 (x + x1) - f2 (x + x1) - (f1 x - f2 x)) / x1 -
   (l1 - l2)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
forall h : R,
(f1 (x + h) - f1 x) / h - (f2 (x + h) - f2 x) / h =
(f1 (x + h) - f2 (x + h) - (f1 x - f2 x)) / h
  assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
H3:forall h : R,
(f1 (x + h) - f1 x) / h - (f2 (x + h) - f2 x) / h =
(f1 (x + h) - f2 (x + h) - (f1 x - f2 x)) / h
H4:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   x1 <> 0 /\ Rabs (x1 - 0) < alp ->
   Rabs
     ((f1 (x + x1) - f1 x) / x1 -
      (f2 (x + x1) - f2 x) / x1 - (l1 - l2)) <
   eps0)
eps:R
H5:eps > 0
x0:R
H6:x0 > 0 /\
(forall x1 : R,
 x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
 Rabs
   ((f1 (x + x1) - f1 x) / x1 -
    (f2 (x + x1) - f2 x) / x1 - (l1 - l2)) < eps)
H7:x0 > 0
H8:forall x1 : R,
x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
Rabs
  ((f1 (x + x1) - f1 x) / x1 -
   (f2 (x + x1) - f2 x) / x1 - (l1 - l2)) < eps
forall x1 : R,
x1 <> 0 /\ Rabs (x1 - 0) < x0 ->
Rabs
  ((f1 (x + x1) - f2 (x + x1) - (f1 x - f2 x)) / x1 -
   (l1 - l2)) < eps
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
forall h : R,
(f1 (x + h) - f1 x) / h - (f2 (x + h) - f2 x) / h =
(f1 (x + h) - f2 (x + h) - (f1 x - f2 x)) / h
  intros; rewrite <- H3; apply H8; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:limit1_in (fun h : R => (f1 (x + h) - f1 x) / h)
  (fun h : R => h <> 0) l1 0
H2:limit1_in (fun h : R => (f2 (x + h) - f2 x) / h)
  (fun h : R => h <> 0) l2 0
forall h : R,
(f1 (x + h) - f1 x) / h - (f2 (x + h) - f2 x) / h =
(f1 (x + h) - f2 (x + h) - (f1 x - f2 x)) / h
  intro; unfold Rdiv; ring.
Qed.

Lemma derivable_pt_lim_mult :
  forall f1 f2 (x l1 l2:R),
    derivable_pt_lim f1 x l1 ->
    derivable_pt_lim f2 x l2 ->
    derivable_pt_lim (f1 * f2) x (l1 * f2 x + f1 x * l2).forall (f1 f2 : R -> R) (x l1 l2 : R),
derivable_pt_lim f1 x l1 ->
derivable_pt_lim f2 x l2 ->
derivable_pt_lim (f1 * f2) x (l1 * f2 x + f1 x * l2)
Proof.forall (f1 f2 : R -> R) (x l1 l2 : R),
derivable_pt_lim f1 x l1 ->
derivable_pt_lim f2 x l2 ->
derivable_pt_lim (f1 * f2) x (l1 * f2 x + f1 x * l2)
  intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
derivable_pt_lim (f1 * f2) x (l1 * f2 x + f1 x * l2)
  assert (H1 := derivable_pt_lim_D_in f1 (fun y:R => l1) x).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:D_in f1 (fun _ : R => l1) no_cond x <->
derivable_pt_lim f1 x ((fun _ : R => l1) x)
derivable_pt_lim (f1 * f2) x (l1 * f2 x + f1 x * l2)
  elim H1; intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:D_in f1 (fun _ : R => l1) no_cond x <->
derivable_pt_lim f1 x ((fun _ : R => l1) x)
H2:D_in f1 (fun _ : R => l1) no_cond x ->
derivable_pt_lim f1 x l1
H3:derivable_pt_lim f1 x l1 ->
D_in f1 (fun _ : R => l1) no_cond x
derivable_pt_lim (f1 * f2) x (l1 * f2 x + f1 x * l2)
  assert (H4 := H3 H).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:D_in f1 (fun _ : R => l1) no_cond x <->
derivable_pt_lim f1 x ((fun _ : R => l1) x)
H2:D_in f1 (fun _ : R => l1) no_cond x ->
derivable_pt_lim f1 x l1
H3:derivable_pt_lim f1 x l1 ->
D_in f1 (fun _ : R => l1) no_cond x
H4:D_in f1 (fun _ : R => l1) no_cond x
derivable_pt_lim (f1 * f2) x (l1 * f2 x + f1 x * l2)
  assert (H5 := derivable_pt_lim_D_in f2 (fun y:R => l2) x).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:D_in f1 (fun _ : R => l1) no_cond x <->
derivable_pt_lim f1 x ((fun _ : R => l1) x)
H2:D_in f1 (fun _ : R => l1) no_cond x ->
derivable_pt_lim f1 x l1
H3:derivable_pt_lim f1 x l1 ->
D_in f1 (fun _ : R => l1) no_cond x
H4:D_in f1 (fun _ : R => l1) no_cond x
H5:D_in f2 (fun _ : R => l2) no_cond x <->
derivable_pt_lim f2 x ((fun _ : R => l2) x)
derivable_pt_lim (f1 * f2) x (l1 * f2 x + f1 x * l2)
  elim H5; intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:D_in f1 (fun _ : R => l1) no_cond x <->
derivable_pt_lim f1 x ((fun _ : R => l1) x)
H2:D_in f1 (fun _ : R => l1) no_cond x ->
derivable_pt_lim f1 x l1
H3:derivable_pt_lim f1 x l1 ->
D_in f1 (fun _ : R => l1) no_cond x
H4:D_in f1 (fun _ : R => l1) no_cond x
H5:D_in f2 (fun _ : R => l2) no_cond x <->
derivable_pt_lim f2 x ((fun _ : R => l2) x)
H6:D_in f2 (fun _ : R => l2) no_cond x ->
derivable_pt_lim f2 x l2
H7:derivable_pt_lim f2 x l2 ->
D_in f2 (fun _ : R => l2) no_cond x
derivable_pt_lim (f1 * f2) x (l1 * f2 x + f1 x * l2)
  assert (H8 := H7 H0).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H1:D_in f1 (fun _ : R => l1) no_cond x <->
derivable_pt_lim f1 x ((fun _ : R => l1) x)
H2:D_in f1 (fun _ : R => l1) no_cond x ->
derivable_pt_lim f1 x l1
H3:derivable_pt_lim f1 x l1 ->
D_in f1 (fun _ : R => l1) no_cond x
H4:D_in f1 (fun _ : R => l1) no_cond x
H5:D_in f2 (fun _ : R => l2) no_cond x <->
derivable_pt_lim f2 x ((fun _ : R => l2) x)
H6:D_in f2 (fun _ : R => l2) no_cond x ->
derivable_pt_lim f2 x l2
H7:derivable_pt_lim f2 x l2 ->
D_in f2 (fun _ : R => l2) no_cond x
H8:D_in f2 (fun _ : R => l2) no_cond x
derivable_pt_lim (f1 * f2) x (l1 * f2 x + f1 x * l2)
  clear H1 H2 H3 H5 H6 H7.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H4:D_in f1 (fun _ : R => l1) no_cond x
H8:D_in f2 (fun _ : R => l2) no_cond x
derivable_pt_lim (f1 * f2) x (l1 * f2 x + f1 x * l2)
  assert
    (H1 :=
      derivable_pt_lim_D_in (f1 * f2)%F (fun y:R => l1 * f2 x + f1 x * l2) x).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H4:D_in f1 (fun _ : R => l1) no_cond x
H8:D_in f2 (fun _ : R => l2) no_cond x
H1:D_in (f1 * f2)%F
  (fun _ : R => l1 * f2 x + f1 x * l2) no_cond x <->
derivable_pt_lim (f1 * f2) x
  ((fun _ : R => l1 * f2 x + f1 x * l2) x)
derivable_pt_lim (f1 * f2) x (l1 * f2 x + f1 x * l2)
  elim H1; intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H4:D_in f1 (fun _ : R => l1) no_cond x
H8:D_in f2 (fun _ : R => l2) no_cond x
H1:D_in (f1 * f2)%F
  (fun _ : R => l1 * f2 x + f1 x * l2) no_cond x <->
derivable_pt_lim (f1 * f2) x
  ((fun _ : R => l1 * f2 x + f1 x * l2) x)
H2:D_in (f1 * f2)%F
  (fun _ : R => l1 * f2 x + f1 x * l2) no_cond x ->
derivable_pt_lim (f1 * f2) x
  (l1 * f2 x + f1 x * l2)
H3:derivable_pt_lim (f1 * f2) x
  (l1 * f2 x + f1 x * l2) ->
D_in (f1 * f2)%F
  (fun _ : R => l1 * f2 x + f1 x * l2) no_cond x
derivable_pt_lim (f1 * f2) x (l1 * f2 x + f1 x * l2)
  clear H1 H3.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H4:D_in f1 (fun _ : R => l1) no_cond x
H8:D_in f2 (fun _ : R => l2) no_cond x
H2:D_in (f1 * f2)%F
  (fun _ : R => l1 * f2 x + f1 x * l2) no_cond x ->
derivable_pt_lim (f1 * f2) x
  (l1 * f2 x + f1 x * l2)
derivable_pt_lim (f1 * f2) x (l1 * f2 x + f1 x * l2)
  apply H2.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H4:D_in f1 (fun _ : R => l1) no_cond x
H8:D_in f2 (fun _ : R => l2) no_cond x
H2:D_in (f1 * f2)%F
  (fun _ : R => l1 * f2 x + f1 x * l2) no_cond x ->
derivable_pt_lim (f1 * f2) x
  (l1 * f2 x + f1 x * l2)
D_in (f1 * f2)%F (fun _ : R => l1 * f2 x + f1 x * l2)
  no_cond x
  unfold mult_fct.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 x l2
H4:D_in f1 (fun _ : R => l1) no_cond x
H8:D_in f2 (fun _ : R => l2) no_cond x
H2:D_in (f1 * f2)%F
  (fun _ : R => l1 * f2 x + f1 x * l2) no_cond x ->
derivable_pt_lim (f1 * f2) x
  (l1 * f2 x + f1 x * l2)
D_in (fun x0 : R => f1 x0 * f2 x0)
  (fun _ : R => l1 * f2 x + f1 x * l2) no_cond x
  apply (Dmult no_cond (fun y:R => l1) (fun y:R => l2) f1 f2 x); assumption.
Qed.

Lemma derivable_pt_lim_const : forall a x:R, derivable_pt_lim (fct_cte a) x 0.forall a x : R, derivable_pt_lim (fct_cte a) x 0
Proof.forall a x : R, derivable_pt_lim (fct_cte a) x 0
  intros; unfold fct_cte, derivable_pt_lim.a, x:R
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta -> Rabs ((a - a) / h - 0) < eps
  intros; exists (mkposreal 1 Rlt_0_1); intros; unfold Rminus;
    rewrite Rplus_opp_r; unfold Rdiv; rewrite Rmult_0_l;
      rewrite Rplus_opp_r; rewrite Rabs_R0; assumption.
Qed.

Lemma derivable_pt_lim_scal :
  forall f (a x l:R),
    derivable_pt_lim f x l -> derivable_pt_lim (mult_real_fct a f) x (a * l).forall (f : R -> R) (a x l : R),
derivable_pt_lim f x l ->
derivable_pt_lim (mult_real_fct a f) x (a * l)
Proof.forall (f : R -> R) (a x l : R),
derivable_pt_lim f x l ->
derivable_pt_lim (mult_real_fct a f) x (a * l)
  intros.f:R -> R
a, x, l:R
H:derivable_pt_lim f x l
derivable_pt_lim (mult_real_fct a f) x (a * l)
  assert (H0 := derivable_pt_lim_const a x).f:R -> R
a, x, l:R
H:derivable_pt_lim f x l
H0:derivable_pt_lim (fct_cte a) x 0
derivable_pt_lim (mult_real_fct a f) x (a * l)
  replace (mult_real_fct a f) with (fct_cte a * f)%F.f:R -> R
a, x, l:R
H:derivable_pt_lim f x l
H0:derivable_pt_lim (fct_cte a) x 0
derivable_pt_lim (fct_cte a * f) x (a * l)
f:R -> R
a, x, l:R
H:derivable_pt_lim f x l
H0:derivable_pt_lim (fct_cte a) x 0
(fct_cte a * f)%F = mult_real_fct a f
  replace (a * l) with (0 * f x + a * l); [ idtac | ring ].f:R -> R
a, x, l:R
H:derivable_pt_lim f x l
H0:derivable_pt_lim (fct_cte a) x 0
derivable_pt_lim (fct_cte a * f) x (0 * f x + a * l)
f:R -> R
a, x, l:R
H:derivable_pt_lim f x l
H0:derivable_pt_lim (fct_cte a) x 0
(fct_cte a * f)%F = mult_real_fct a f
  apply (derivable_pt_lim_mult (fct_cte a) f x 0 l); assumption.f:R -> R
a, x, l:R
H:derivable_pt_lim f x l
H0:derivable_pt_lim (fct_cte a) x 0
(fct_cte a * f)%F = mult_real_fct a f
  unfold mult_real_fct, mult_fct, fct_cte; reflexivity.
Qed.

Lemma derivable_pt_lim_div_scal :
  forall f x l a, derivable_pt_lim f x l ->
     derivable_pt_lim (fun y => f y / a) x (l / a).forall (f : R -> R) (x l a : R),
derivable_pt_lim f x l ->
derivable_pt_lim (fun y : R => f y / a) x (l / a)
intros f x l a df;
  apply (derivable_pt_lim_ext (fun y => /a * f y)).f:R -> R
x, l, a:R
df:derivable_pt_lim f x l
forall z : R, / a * f z = f z / a
f:R -> R
x, l, a:R
df:derivable_pt_lim f x l
derivable_pt_lim (fun y : R => / a * f y) x (l / a)
 intros z; rewrite Rmult_comm; reflexivity.f:R -> R
x, l, a:R
df:derivable_pt_lim f x l
derivable_pt_lim (fun y : R => / a * f y) x (l / a)
unfold Rdiv; rewrite Rmult_comm; apply derivable_pt_lim_scal; assumption.
Qed.

Lemma derivable_pt_lim_scal_right :
  forall f x l a, derivable_pt_lim f x l ->
     derivable_pt_lim (fun y => f y * a) x (l * a).forall (f : R -> R) (x l a : R),
derivable_pt_lim f x l ->
derivable_pt_lim (fun y : R => f y * a) x (l * a)
intros f x l a df;
  apply (derivable_pt_lim_ext (fun y => a * f y)).f:R -> R
x, l, a:R
df:derivable_pt_lim f x l
forall z : R, a * f z = f z * a
f:R -> R
x, l, a:R
df:derivable_pt_lim f x l
derivable_pt_lim (fun y : R => a * f y) x (l * a)
 intros z; rewrite Rmult_comm; reflexivity.f:R -> R
x, l, a:R
df:derivable_pt_lim f x l
derivable_pt_lim (fun y : R => a * f y) x (l * a)
unfold Rdiv; rewrite Rmult_comm; apply derivable_pt_lim_scal; assumption.
Qed.

Lemma derivable_pt_lim_id : forall x:R, derivable_pt_lim id x 1.forall x : R, derivable_pt_lim id x 1
Proof.forall x : R, derivable_pt_lim id x 1
  intro; unfold derivable_pt_lim.x:R
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((id (x + h) - id x) / h - 1) < eps
  intros eps Heps; exists (mkposreal eps Heps); intros h H1 H2;
    unfold id; replace ((x + h - x) / h - 1) with 0.x, eps:R
Heps:0 < eps
h:R
H1:h <> 0
H2:Rabs h < {| pos := eps; cond_pos := Heps |}
Rabs 0 < eps
x, eps:R
Heps:0 < eps
h:R
H1:h <> 0
H2:Rabs h < {| pos := eps; cond_pos := Heps |}
0 = (x + h - x) / h - 1
  rewrite Rabs_R0; apply Rle_lt_trans with (Rabs h).x, eps:R
Heps:0 < eps
h:R
H1:h <> 0
H2:Rabs h < {| pos := eps; cond_pos := Heps |}
0 <= Rabs h
x, eps:R
Heps:0 < eps
h:R
H1:h <> 0
H2:Rabs h < {| pos := eps; cond_pos := Heps |}
Rabs h < eps
x, eps:R
Heps:0 < eps
h:R
H1:h <> 0
H2:Rabs h < {| pos := eps; cond_pos := Heps |}
0 = (x + h - x) / h - 1
  apply Rabs_pos.x, eps:R
Heps:0 < eps
h:R
H1:h <> 0
H2:Rabs h < {| pos := eps; cond_pos := Heps |}
Rabs h < eps
x, eps:R
Heps:0 < eps
h:R
H1:h <> 0
H2:Rabs h < {| pos := eps; cond_pos := Heps |}
0 = (x + h - x) / h - 1
  assumption.x, eps:R
Heps:0 < eps
h:R
H1:h <> 0
H2:Rabs h < {| pos := eps; cond_pos := Heps |}
0 = (x + h - x) / h - 1
  unfold Rminus; rewrite Rplus_assoc; rewrite (Rplus_comm x);
    rewrite Rplus_assoc.x, eps:R
Heps:0 < eps
h:R
H1:h <> 0
H2:Rabs h < {| pos := eps; cond_pos := Heps |}
0 = (h + (- x + x)) / h + - (1)
  rewrite Rplus_opp_l; rewrite Rplus_0_r; unfold Rdiv;
    rewrite <- Rinv_r_sym.x, eps:R
Heps:0 < eps
h:R
H1:h <> 0
H2:Rabs h < {| pos := eps; cond_pos := Heps |}
0 = 1 + - (1)
x, eps:R
Heps:0 < eps
h:R
H1:h <> 0
H2:Rabs h < {| pos := eps; cond_pos := Heps |}
h <> 0
  symmetry ; apply Rplus_opp_r.x, eps:R
Heps:0 < eps
h:R
H1:h <> 0
H2:Rabs h < {| pos := eps; cond_pos := Heps |}
h <> 0
  assumption.
Qed.

Lemma derivable_pt_lim_Rsqr : forall x:R, derivable_pt_lim Rsqr x (2 * x).forall x : R, derivable_pt_lim Rsqr x (2 * x)
Proof.forall x : R, derivable_pt_lim Rsqr x (2 * x)
  intro; unfold derivable_pt_lim.x:R
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs (((x + h)² - x²) / h - 2 * x) < eps
  unfold Rsqr; intros eps Heps; exists (mkposreal eps Heps);
    intros h H1 H2; replace (((x + h) * (x + h) - x * x) / h - 2 * x) with h.x, eps:R
Heps:0 < eps
h:R
H1:h <> 0
H2:Rabs h < {| pos := eps; cond_pos := Heps |}
Rabs h < eps
x, eps:R
Heps:0 < eps
h:R
H1:h <> 0
H2:Rabs h < {| pos := eps; cond_pos := Heps |}
h = ((x + h) * (x + h) - x * x) / h - 2 * x
  assumption.x, eps:R
Heps:0 < eps
h:R
H1:h <> 0
H2:Rabs h < {| pos := eps; cond_pos := Heps |}
h = ((x + h) * (x + h) - x * x) / h - 2 * x
  replace ((x + h) * (x + h) - x * x) with (2 * x * h + h * h);
  [ idtac | ring ].x, eps:R
Heps:0 < eps
h:R
H1:h <> 0
H2:Rabs h < {| pos := eps; cond_pos := Heps |}
h = (2 * x * h + h * h) / h - 2 * x
  unfold Rdiv; rewrite Rmult_plus_distr_r.x, eps:R
Heps:0 < eps
h:R
H1:h <> 0
H2:Rabs h < {| pos := eps; cond_pos := Heps |}
h = 2 * x * h * / h + h * h * / h - 2 * x
  repeat rewrite Rmult_assoc.x, eps:R
Heps:0 < eps
h:R
H1:h <> 0
H2:Rabs h < {| pos := eps; cond_pos := Heps |}
h = 2 * (x * (h * / h)) + h * (h * / h) - 2 * x
  repeat rewrite <- Rinv_r_sym; [ idtac | assumption ].x, eps:R
Heps:0 < eps
h:R
H1:h <> 0
H2:Rabs h < {| pos := eps; cond_pos := Heps |}
h = 2 * (x * 1) + h * 1 - 2 * x
  ring.
Qed.

Lemma derivable_pt_lim_comp :
  forall f1 f2 (x l1 l2:R),
    derivable_pt_lim f1 x l1 ->
    derivable_pt_lim f2 (f1 x) l2 -> derivable_pt_lim (f2 o f1) x (l2 * l1).forall (f1 f2 : R -> R) (x l1 l2 : R),
derivable_pt_lim f1 x l1 ->
derivable_pt_lim f2 (f1 x) l2 ->
derivable_pt_lim (f2 o f1) x (l2 * l1)
Proof.forall (f1 f2 : R -> R) (x l1 l2 : R),
derivable_pt_lim f1 x l1 ->
derivable_pt_lim f2 (f1 x) l2 ->
derivable_pt_lim (f2 o f1) x (l2 * l1)
  intros; assert (H1 := derivable_pt_lim_D_in f1 (fun y:R => l1) x).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 (f1 x) l2
H1:D_in f1 (fun _ : R => l1) no_cond x <->
derivable_pt_lim f1 x ((fun _ : R => l1) x)
derivable_pt_lim (f2 o f1) x (l2 * l1)
  elim H1; intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 (f1 x) l2
H1:D_in f1 (fun _ : R => l1) no_cond x <->
derivable_pt_lim f1 x ((fun _ : R => l1) x)
H2:D_in f1 (fun _ : R => l1) no_cond x ->
derivable_pt_lim f1 x l1
H3:derivable_pt_lim f1 x l1 ->
D_in f1 (fun _ : R => l1) no_cond x
derivable_pt_lim (f2 o f1) x (l2 * l1)
  assert (H4 := H3 H).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 (f1 x) l2
H1:D_in f1 (fun _ : R => l1) no_cond x <->
derivable_pt_lim f1 x ((fun _ : R => l1) x)
H2:D_in f1 (fun _ : R => l1) no_cond x ->
derivable_pt_lim f1 x l1
H3:derivable_pt_lim f1 x l1 ->
D_in f1 (fun _ : R => l1) no_cond x
H4:D_in f1 (fun _ : R => l1) no_cond x
derivable_pt_lim (f2 o f1) x (l2 * l1)
  assert (H5 := derivable_pt_lim_D_in f2 (fun y:R => l2) (f1 x)).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 (f1 x) l2
H1:D_in f1 (fun _ : R => l1) no_cond x <->
derivable_pt_lim f1 x ((fun _ : R => l1) x)
H2:D_in f1 (fun _ : R => l1) no_cond x ->
derivable_pt_lim f1 x l1
H3:derivable_pt_lim f1 x l1 ->
D_in f1 (fun _ : R => l1) no_cond x
H4:D_in f1 (fun _ : R => l1) no_cond x
H5:D_in f2 (fun _ : R => l2) no_cond (f1 x) <->
derivable_pt_lim f2 (f1 x)
  ((fun _ : R => l2) (f1 x))
derivable_pt_lim (f2 o f1) x (l2 * l1)
  elim H5; intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 (f1 x) l2
H1:D_in f1 (fun _ : R => l1) no_cond x <->
derivable_pt_lim f1 x ((fun _ : R => l1) x)
H2:D_in f1 (fun _ : R => l1) no_cond x ->
derivable_pt_lim f1 x l1
H3:derivable_pt_lim f1 x l1 ->
D_in f1 (fun _ : R => l1) no_cond x
H4:D_in f1 (fun _ : R => l1) no_cond x
H5:D_in f2 (fun _ : R => l2) no_cond (f1 x) <->
derivable_pt_lim f2 (f1 x)
  ((fun _ : R => l2) (f1 x))
H6:D_in f2 (fun _ : R => l2) no_cond (f1 x) ->
derivable_pt_lim f2 (f1 x) l2
H7:derivable_pt_lim f2 (f1 x) l2 ->
D_in f2 (fun _ : R => l2) no_cond (f1 x)
derivable_pt_lim (f2 o f1) x (l2 * l1)
  assert (H8 := H7 H0).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 (f1 x) l2
H1:D_in f1 (fun _ : R => l1) no_cond x <->
derivable_pt_lim f1 x ((fun _ : R => l1) x)
H2:D_in f1 (fun _ : R => l1) no_cond x ->
derivable_pt_lim f1 x l1
H3:derivable_pt_lim f1 x l1 ->
D_in f1 (fun _ : R => l1) no_cond x
H4:D_in f1 (fun _ : R => l1) no_cond x
H5:D_in f2 (fun _ : R => l2) no_cond (f1 x) <->
derivable_pt_lim f2 (f1 x)
  ((fun _ : R => l2) (f1 x))
H6:D_in f2 (fun _ : R => l2) no_cond (f1 x) ->
derivable_pt_lim f2 (f1 x) l2
H7:derivable_pt_lim f2 (f1 x) l2 ->
D_in f2 (fun _ : R => l2) no_cond (f1 x)
H8:D_in f2 (fun _ : R => l2) no_cond (f1 x)
derivable_pt_lim (f2 o f1) x (l2 * l1)
  clear H1 H2 H3 H5 H6 H7.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 (f1 x) l2
H4:D_in f1 (fun _ : R => l1) no_cond x
H8:D_in f2 (fun _ : R => l2) no_cond (f1 x)
derivable_pt_lim (f2 o f1) x (l2 * l1)
  assert (H1 := derivable_pt_lim_D_in (f2 o f1)%F (fun y:R => l2 * l1) x).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 (f1 x) l2
H4:D_in f1 (fun _ : R => l1) no_cond x
H8:D_in f2 (fun _ : R => l2) no_cond (f1 x)
H1:D_in (f2 o f1)%F (fun _ : R => l2 * l1) no_cond x <->
derivable_pt_lim (f2 o f1) x
  ((fun _ : R => l2 * l1) x)
derivable_pt_lim (f2 o f1) x (l2 * l1)
  elim H1; intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 (f1 x) l2
H4:D_in f1 (fun _ : R => l1) no_cond x
H8:D_in f2 (fun _ : R => l2) no_cond (f1 x)
H1:D_in (f2 o f1)%F (fun _ : R => l2 * l1) no_cond x <->
derivable_pt_lim (f2 o f1) x
  ((fun _ : R => l2 * l1) x)
H2:D_in (f2 o f1)%F (fun _ : R => l2 * l1) no_cond x ->
derivable_pt_lim (f2 o f1) x (l2 * l1)
H3:derivable_pt_lim (f2 o f1) x (l2 * l1) ->
D_in (f2 o f1)%F (fun _ : R => l2 * l1) no_cond x
derivable_pt_lim (f2 o f1) x (l2 * l1)
  clear H1 H3; apply H2.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 (f1 x) l2
H4:D_in f1 (fun _ : R => l1) no_cond x
H8:D_in f2 (fun _ : R => l2) no_cond (f1 x)
H2:D_in (f2 o f1)%F (fun _ : R => l2 * l1) no_cond x ->
derivable_pt_lim (f2 o f1) x (l2 * l1)
D_in (f2 o f1)%F (fun _ : R => l2 * l1) no_cond x
  unfold comp;
    cut
      (D_in (fun x0:R => f2 (f1 x0)) (fun y:R => l2 * l1)
        (Dgf no_cond no_cond f1) x ->
        D_in (fun x0:R => f2 (f1 x0)) (fun y:R => l2 * l1) no_cond x).f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 (f1 x) l2
H4:D_in f1 (fun _ : R => l1) no_cond x
H8:D_in f2 (fun _ : R => l2) no_cond (f1 x)
H2:D_in (f2 o f1)%F (fun _ : R => l2 * l1) no_cond x ->
derivable_pt_lim (f2 o f1) x (l2 * l1)
(D_in (fun x0 : R => f2 (f1 x0))
   (fun _ : R => l2 * l1) (Dgf no_cond no_cond f1) x ->
 D_in (fun x0 : R => f2 (f1 x0))
   (fun _ : R => l2 * l1) no_cond x) ->
D_in (fun x0 : R => f2 (f1 x0)) (fun _ : R => l2 * l1)
  no_cond x
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 (f1 x) l2
H4:D_in f1 (fun _ : R => l1) no_cond x
H8:D_in f2 (fun _ : R => l2) no_cond (f1 x)
H2:D_in (f2 o f1)%F (fun _ : R => l2 * l1) no_cond x ->
derivable_pt_lim (f2 o f1) x (l2 * l1)
D_in (fun x0 : R => f2 (f1 x0)) (fun _ : R => l2 * l1)
  (Dgf no_cond no_cond f1) x ->
D_in (fun x0 : R => f2 (f1 x0)) (fun _ : R => l2 * l1)
  no_cond x
  intro; apply H1.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 (f1 x) l2
H4:D_in f1 (fun _ : R => l1) no_cond x
H8:D_in f2 (fun _ : R => l2) no_cond (f1 x)
H2:D_in (f2 o f1)%F (fun _ : R => l2 * l1) no_cond x ->
derivable_pt_lim (f2 o f1) x (l2 * l1)
H1:D_in (fun x0 : R => f2 (f1 x0))
  (fun _ : R => l2 * l1) (Dgf no_cond no_cond f1)
  x ->
D_in (fun x0 : R => f2 (f1 x0))
  (fun _ : R => l2 * l1) no_cond x
D_in (fun x0 : R => f2 (f1 x0)) (fun _ : R => l2 * l1)
  (Dgf no_cond no_cond f1) x
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 (f1 x) l2
H4:D_in f1 (fun _ : R => l1) no_cond x
H8:D_in f2 (fun _ : R => l2) no_cond (f1 x)
H2:D_in (f2 o f1)%F (fun _ : R => l2 * l1) no_cond x ->
derivable_pt_lim (f2 o f1) x (l2 * l1)
D_in (fun x0 : R => f2 (f1 x0)) (fun _ : R => l2 * l1)
  (Dgf no_cond no_cond f1) x ->
D_in (fun x0 : R => f2 (f1 x0)) (fun _ : R => l2 * l1)
  no_cond x
  rewrite Rmult_comm;
    apply (Dcomp no_cond no_cond (fun y:R => l1) (fun y:R => l2) f1 f2 x);
      assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 (f1 x) l2
H4:D_in f1 (fun _ : R => l1) no_cond x
H8:D_in f2 (fun _ : R => l2) no_cond (f1 x)
H2:D_in (f2 o f1)%F (fun _ : R => l2 * l1) no_cond x ->
derivable_pt_lim (f2 o f1) x (l2 * l1)
D_in (fun x0 : R => f2 (f1 x0)) (fun _ : R => l2 * l1)
  (Dgf no_cond no_cond f1) x ->
D_in (fun x0 : R => f2 (f1 x0)) (fun _ : R => l2 * l1)
  no_cond x
  unfold Dgf, D_in, no_cond; unfold limit1_in;
    unfold limit_in; unfold dist; simpl;
      unfold R_dist; intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 (f1 x) l2
H4:D_in f1 (fun _ : R => l1) no_cond x
H8:D_in f2 (fun _ : R => l2) no_cond (f1 x)
H2:D_in (f2 o f1)%F (fun _ : R => l2 * l1) no_cond x ->
derivable_pt_lim (f2 o f1) x (l2 * l1)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x (fun _ : R => True /\ True) x x0 /\
   Rabs (x0 - x) < alp ->
   Rabs
     ((f2 (f1 x0) - f2 (f1 x)) / (x0 - x) -
      l2 * l1) < eps0)
eps:R
H3:eps > 0
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x (fun _ : R => True) x x0 /\ Rabs (x0 - x) < alp ->
   Rabs
     ((f2 (f1 x0) - f2 (f1 x)) / (x0 - x) - l2 * l1) <
   eps)
  elim (H1 eps H3); intros.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 (f1 x) l2
H4:D_in f1 (fun _ : R => l1) no_cond x
H8:D_in f2 (fun _ : R => l2) no_cond (f1 x)
H2:D_in (f2 o f1)%F (fun _ : R => l2 * l1) no_cond x ->
derivable_pt_lim (f2 o f1) x (l2 * l1)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x (fun _ : R => True /\ True) x x1 /\
   Rabs (x1 - x) < alp ->
   Rabs
     ((f2 (f1 x1) - f2 (f1 x)) / (x1 - x) -
      l2 * l1) < eps0)
eps:R
H3:eps > 0
x0:R
H5:x0 > 0 /\
(forall x1 : R,
 D_x (fun _ : R => True /\ True) x x1 /\
 Rabs (x1 - x) < x0 ->
 Rabs
   ((f2 (f1 x1) - f2 (f1 x)) / (x1 - x) - l2 * l1) <
 eps)
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x (fun _ : R => True) x x1 /\ Rabs (x1 - x) < alp ->
   Rabs
     ((f2 (f1 x1) - f2 (f1 x)) / (x1 - x) - l2 * l1) <
   eps)
  exists x0; intros; split.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 (f1 x) l2
H4:D_in f1 (fun _ : R => l1) no_cond x
H8:D_in f2 (fun _ : R => l2) no_cond (f1 x)
H2:D_in (f2 o f1)%F (fun _ : R => l2 * l1) no_cond x ->
derivable_pt_lim (f2 o f1) x (l2 * l1)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x (fun _ : R => True /\ True) x x1 /\
   Rabs (x1 - x) < alp ->
   Rabs
     ((f2 (f1 x1) - f2 (f1 x)) / (x1 - x) -
      l2 * l1) < eps0)
eps:R
H3:eps > 0
x0:R
H5:x0 > 0 /\
(forall x1 : R,
 D_x (fun _ : R => True /\ True) x x1 /\
 Rabs (x1 - x) < x0 ->
 Rabs
   ((f2 (f1 x1) - f2 (f1 x)) / (x1 - x) - l2 * l1) <
 eps)
x0 > 0
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 (f1 x) l2
H4:D_in f1 (fun _ : R => l1) no_cond x
H8:D_in f2 (fun _ : R => l2) no_cond (f1 x)
H2:D_in (f2 o f1)%F (fun _ : R => l2 * l1) no_cond x ->
derivable_pt_lim (f2 o f1) x (l2 * l1)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x (fun _ : R => True /\ True) x x1 /\
   Rabs (x1 - x) < alp ->
   Rabs
     ((f2 (f1 x1) - f2 (f1 x)) / (x1 - x) -
      l2 * l1) < eps0)
eps:R
H3:eps > 0
x0:R
H5:x0 > 0 /\
(forall x1 : R,
 D_x (fun _ : R => True /\ True) x x1 /\
 Rabs (x1 - x) < x0 ->
 Rabs
   ((f2 (f1 x1) - f2 (f1 x)) / (x1 - x) - l2 * l1) <
 eps)
forall x1 : R,
D_x (fun _ : R => True) x x1 /\ Rabs (x1 - x) < x0 ->
Rabs ((f2 (f1 x1) - f2 (f1 x)) / (x1 - x) - l2 * l1) <
eps
  elim H5; intros; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 (f1 x) l2
H4:D_in f1 (fun _ : R => l1) no_cond x
H8:D_in f2 (fun _ : R => l2) no_cond (f1 x)
H2:D_in (f2 o f1)%F (fun _ : R => l2 * l1) no_cond x ->
derivable_pt_lim (f2 o f1) x (l2 * l1)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x (fun _ : R => True /\ True) x x1 /\
   Rabs (x1 - x) < alp ->
   Rabs
     ((f2 (f1 x1) - f2 (f1 x)) / (x1 - x) -
      l2 * l1) < eps0)
eps:R
H3:eps > 0
x0:R
H5:x0 > 0 /\
(forall x1 : R,
 D_x (fun _ : R => True /\ True) x x1 /\
 Rabs (x1 - x) < x0 ->
 Rabs
   ((f2 (f1 x1) - f2 (f1 x)) / (x1 - x) - l2 * l1) <
 eps)
forall x1 : R,
D_x (fun _ : R => True) x x1 /\ Rabs (x1 - x) < x0 ->
Rabs ((f2 (f1 x1) - f2 (f1 x)) / (x1 - x) - l2 * l1) <
eps
  intros; elim H5; intros; apply H9; split.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 (f1 x) l2
H4:D_in f1 (fun _ : R => l1) no_cond x
H8:D_in f2 (fun _ : R => l2) no_cond (f1 x)
H2:D_in (f2 o f1)%F (fun _ : R => l2 * l1) no_cond x ->
derivable_pt_lim (f2 o f1) x (l2 * l1)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   D_x (fun _ : R => True /\ True) x x2 /\
   Rabs (x2 - x) < alp ->
   Rabs
     ((f2 (f1 x2) - f2 (f1 x)) / (x2 - x) -
      l2 * l1) < eps0)
eps:R
H3:eps > 0
x0:R
H5:x0 > 0 /\
(forall x2 : R,
 D_x (fun _ : R => True /\ True) x x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs
   ((f2 (f1 x2) - f2 (f1 x)) / (x2 - x) - l2 * l1) <
 eps)
x1:R
H6:D_x (fun _ : R => True) x x1 /\
Rabs (x1 - x) < x0
H7:x0 > 0
H9:forall x2 : R,
D_x (fun _ : R => True /\ True) x x2 /\
Rabs (x2 - x) < x0 ->
Rabs
  ((f2 (f1 x2) - f2 (f1 x)) / (x2 - x) - l2 * l1) <
eps
D_x (fun _ : R => True /\ True) x x1
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 (f1 x) l2
H4:D_in f1 (fun _ : R => l1) no_cond x
H8:D_in f2 (fun _ : R => l2) no_cond (f1 x)
H2:D_in (f2 o f1)%F (fun _ : R => l2 * l1) no_cond x ->
derivable_pt_lim (f2 o f1) x (l2 * l1)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   D_x (fun _ : R => True /\ True) x x2 /\
   Rabs (x2 - x) < alp ->
   Rabs
     ((f2 (f1 x2) - f2 (f1 x)) / (x2 - x) -
      l2 * l1) < eps0)
eps:R
H3:eps > 0
x0:R
H5:x0 > 0 /\
(forall x2 : R,
 D_x (fun _ : R => True /\ True) x x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs
   ((f2 (f1 x2) - f2 (f1 x)) / (x2 - x) - l2 * l1) <
 eps)
x1:R
H6:D_x (fun _ : R => True) x x1 /\
Rabs (x1 - x) < x0
H7:x0 > 0
H9:forall x2 : R,
D_x (fun _ : R => True /\ True) x x2 /\
Rabs (x2 - x) < x0 ->
Rabs
  ((f2 (f1 x2) - f2 (f1 x)) / (x2 - x) - l2 * l1) <
eps
Rabs (x1 - x) < x0
  unfold D_x; split.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 (f1 x) l2
H4:D_in f1 (fun _ : R => l1) no_cond x
H8:D_in f2 (fun _ : R => l2) no_cond (f1 x)
H2:D_in (f2 o f1)%F (fun _ : R => l2 * l1) no_cond x ->
derivable_pt_lim (f2 o f1) x (l2 * l1)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   D_x (fun _ : R => True /\ True) x x2 /\
   Rabs (x2 - x) < alp ->
   Rabs
     ((f2 (f1 x2) - f2 (f1 x)) / (x2 - x) -
      l2 * l1) < eps0)
eps:R
H3:eps > 0
x0:R
H5:x0 > 0 /\
(forall x2 : R,
 D_x (fun _ : R => True /\ True) x x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs
   ((f2 (f1 x2) - f2 (f1 x)) / (x2 - x) - l2 * l1) <
 eps)
x1:R
H6:D_x (fun _ : R => True) x x1 /\
Rabs (x1 - x) < x0
H7:x0 > 0
H9:forall x2 : R,
D_x (fun _ : R => True /\ True) x x2 /\
Rabs (x2 - x) < x0 ->
Rabs
  ((f2 (f1 x2) - f2 (f1 x)) / (x2 - x) - l2 * l1) <
eps
True /\ True
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 (f1 x) l2
H4:D_in f1 (fun _ : R => l1) no_cond x
H8:D_in f2 (fun _ : R => l2) no_cond (f1 x)
H2:D_in (f2 o f1)%F (fun _ : R => l2 * l1) no_cond x ->
derivable_pt_lim (f2 o f1) x (l2 * l1)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   D_x (fun _ : R => True /\ True) x x2 /\
   Rabs (x2 - x) < alp ->
   Rabs
     ((f2 (f1 x2) - f2 (f1 x)) / (x2 - x) -
      l2 * l1) < eps0)
eps:R
H3:eps > 0
x0:R
H5:x0 > 0 /\
(forall x2 : R,
 D_x (fun _ : R => True /\ True) x x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs
   ((f2 (f1 x2) - f2 (f1 x)) / (x2 - x) - l2 * l1) <
 eps)
x1:R
H6:D_x (fun _ : R => True) x x1 /\
Rabs (x1 - x) < x0
H7:x0 > 0
H9:forall x2 : R,
D_x (fun _ : R => True /\ True) x x2 /\
Rabs (x2 - x) < x0 ->
Rabs
  ((f2 (f1 x2) - f2 (f1 x)) / (x2 - x) - l2 * l1) <
eps
x <> x1
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 (f1 x) l2
H4:D_in f1 (fun _ : R => l1) no_cond x
H8:D_in f2 (fun _ : R => l2) no_cond (f1 x)
H2:D_in (f2 o f1)%F (fun _ : R => l2 * l1) no_cond x ->
derivable_pt_lim (f2 o f1) x (l2 * l1)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   D_x (fun _ : R => True /\ True) x x2 /\
   Rabs (x2 - x) < alp ->
   Rabs
     ((f2 (f1 x2) - f2 (f1 x)) / (x2 - x) -
      l2 * l1) < eps0)
eps:R
H3:eps > 0
x0:R
H5:x0 > 0 /\
(forall x2 : R,
 D_x (fun _ : R => True /\ True) x x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs
   ((f2 (f1 x2) - f2 (f1 x)) / (x2 - x) - l2 * l1) <
 eps)
x1:R
H6:D_x (fun _ : R => True) x x1 /\
Rabs (x1 - x) < x0
H7:x0 > 0
H9:forall x2 : R,
D_x (fun _ : R => True /\ True) x x2 /\
Rabs (x2 - x) < x0 ->
Rabs
  ((f2 (f1 x2) - f2 (f1 x)) / (x2 - x) - l2 * l1) <
eps
Rabs (x1 - x) < x0
  split; trivial.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 (f1 x) l2
H4:D_in f1 (fun _ : R => l1) no_cond x
H8:D_in f2 (fun _ : R => l2) no_cond (f1 x)
H2:D_in (f2 o f1)%F (fun _ : R => l2 * l1) no_cond x ->
derivable_pt_lim (f2 o f1) x (l2 * l1)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   D_x (fun _ : R => True /\ True) x x2 /\
   Rabs (x2 - x) < alp ->
   Rabs
     ((f2 (f1 x2) - f2 (f1 x)) / (x2 - x) -
      l2 * l1) < eps0)
eps:R
H3:eps > 0
x0:R
H5:x0 > 0 /\
(forall x2 : R,
 D_x (fun _ : R => True /\ True) x x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs
   ((f2 (f1 x2) - f2 (f1 x)) / (x2 - x) - l2 * l1) <
 eps)
x1:R
H6:D_x (fun _ : R => True) x x1 /\
Rabs (x1 - x) < x0
H7:x0 > 0
H9:forall x2 : R,
D_x (fun _ : R => True /\ True) x x2 /\
Rabs (x2 - x) < x0 ->
Rabs
  ((f2 (f1 x2) - f2 (f1 x)) / (x2 - x) - l2 * l1) <
eps
x <> x1
f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 (f1 x) l2
H4:D_in f1 (fun _ : R => l1) no_cond x
H8:D_in f2 (fun _ : R => l2) no_cond (f1 x)
H2:D_in (f2 o f1)%F (fun _ : R => l2 * l1) no_cond x ->
derivable_pt_lim (f2 o f1) x (l2 * l1)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   D_x (fun _ : R => True /\ True) x x2 /\
   Rabs (x2 - x) < alp ->
   Rabs
     ((f2 (f1 x2) - f2 (f1 x)) / (x2 - x) -
      l2 * l1) < eps0)
eps:R
H3:eps > 0
x0:R
H5:x0 > 0 /\
(forall x2 : R,
 D_x (fun _ : R => True /\ True) x x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs
   ((f2 (f1 x2) - f2 (f1 x)) / (x2 - x) - l2 * l1) <
 eps)
x1:R
H6:D_x (fun _ : R => True) x x1 /\
Rabs (x1 - x) < x0
H7:x0 > 0
H9:forall x2 : R,
D_x (fun _ : R => True /\ True) x x2 /\
Rabs (x2 - x) < x0 ->
Rabs
  ((f2 (f1 x2) - f2 (f1 x)) / (x2 - x) - l2 * l1) <
eps
Rabs (x1 - x) < x0
  elim H6; intros; unfold D_x in H10; elim H10; intros; assumption.f1, f2:R -> R
x, l1, l2:R
H:derivable_pt_lim f1 x l1
H0:derivable_pt_lim f2 (f1 x) l2
H4:D_in f1 (fun _ : R => l1) no_cond x
H8:D_in f2 (fun _ : R => l2) no_cond (f1 x)
H2:D_in (f2 o f1)%F (fun _ : R => l2 * l1) no_cond x ->
derivable_pt_lim (f2 o f1) x (l2 * l1)
H1:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x2 : R,
   D_x (fun _ : R => True /\ True) x x2 /\
   Rabs (x2 - x) < alp ->
   Rabs
     ((f2 (f1 x2) - f2 (f1 x)) / (x2 - x) -
      l2 * l1) < eps0)
eps:R
H3:eps > 0
x0:R
H5:x0 > 0 /\
(forall x2 : R,
 D_x (fun _ : R => True /\ True) x x2 /\
 Rabs (x2 - x) < x0 ->
 Rabs
   ((f2 (f1 x2) - f2 (f1 x)) / (x2 - x) - l2 * l1) <
 eps)
x1:R
H6:D_x (fun _ : R => True) x x1 /\
Rabs (x1 - x) < x0
H7:x0 > 0
H9:forall x2 : R,
D_x (fun _ : R => True /\ True) x x2 /\
Rabs (x2 - x) < x0 ->
Rabs
  ((f2 (f1 x2) - f2 (f1 x)) / (x2 - x) - l2 * l1) <
eps
Rabs (x1 - x) < x0
  elim H6; intros; assumption.
Qed.

Lemma derivable_pt_plus :
  forall f1 f2 (x:R),
    derivable_pt f1 x -> derivable_pt f2 x -> derivable_pt (f1 + f2) x.forall (f1 f2 : R -> R) (x : R),
derivable_pt f1 x ->
derivable_pt f2 x -> derivable_pt (f1 + f2) x
Proof.forall (f1 f2 : R -> R) (x : R),
derivable_pt f1 x ->
derivable_pt f2 x -> derivable_pt (f1 + f2) x
  unfold derivable_pt; intros f1 f2 x X X0.f1, f2:R -> R
x:R
X:{l : R | derivable_pt_abs f1 x l}
X0:{l : R | derivable_pt_abs f2 x l}
{l : R | derivable_pt_abs (f1 + f2) x l}
  elim X; intros.f1, f2:R -> R
x:R
X:{l : R | derivable_pt_abs f1 x l}
X0:{l : R | derivable_pt_abs f2 x l}
x0:R
p:derivable_pt_abs f1 x x0
{l : R | derivable_pt_abs (f1 + f2) x l}
  elim X0; intros.f1, f2:R -> R
x:R
X:{l : R | derivable_pt_abs f1 x l}
X0:{l : R | derivable_pt_abs f2 x l}
x0:R
p:derivable_pt_abs f1 x x0
x1:R
p0:derivable_pt_abs f2 x x1
{l : R | derivable_pt_abs (f1 + f2) x l}
  exists (x0 + x1).f1, f2:R -> R
x:R
X:{l : R | derivable_pt_abs f1 x l}
X0:{l : R | derivable_pt_abs f2 x l}
x0:R
p:derivable_pt_abs f1 x x0
x1:R
p0:derivable_pt_abs f2 x x1
derivable_pt_abs (f1 + f2) x (x0 + x1)
  apply derivable_pt_lim_plus; assumption.
Qed.

Lemma derivable_pt_opp :
  forall f (x:R), derivable_pt f x -> derivable_pt (- f) x.forall (f : R -> R) (x : R),
derivable_pt f x -> derivable_pt (- f) x
Proof.forall (f : R -> R) (x : R),
derivable_pt f x -> derivable_pt (- f) x
  unfold derivable_pt; intros f x X.f:R -> R
x:R
X:{l : R | derivable_pt_abs f x l}
{l : R | derivable_pt_abs (- f) x l}
  elim X; intros.f:R -> R
x:R
X:{l : R | derivable_pt_abs f x l}
x0:R
p:derivable_pt_abs f x x0
{l : R | derivable_pt_abs (- f) x l}
  exists (- x0).f:R -> R
x:R
X:{l : R | derivable_pt_abs f x l}
x0:R
p:derivable_pt_abs f x x0
derivable_pt_abs (- f) x (- x0)
  apply derivable_pt_lim_opp; assumption.
Qed.

Lemma derivable_pt_minus :
  forall f1 f2 (x:R),
    derivable_pt f1 x -> derivable_pt f2 x -> derivable_pt (f1 - f2) x.forall (f1 f2 : R -> R) (x : R),
derivable_pt f1 x ->
derivable_pt f2 x -> derivable_pt (f1 - f2) x
Proof.forall (f1 f2 : R -> R) (x : R),
derivable_pt f1 x ->
derivable_pt f2 x -> derivable_pt (f1 - f2) x
  unfold derivable_pt; intros f1 f2 x X X0.f1, f2:R -> R
x:R
X:{l : R | derivable_pt_abs f1 x l}
X0:{l : R | derivable_pt_abs f2 x l}
{l : R | derivable_pt_abs (f1 - f2) x l}
  elim X; intros.f1, f2:R -> R
x:R
X:{l : R | derivable_pt_abs f1 x l}
X0:{l : R | derivable_pt_abs f2 x l}
x0:R
p:derivable_pt_abs f1 x x0
{l : R | derivable_pt_abs (f1 - f2) x l}
  elim X0; intros.f1, f2:R -> R
x:R
X:{l : R | derivable_pt_abs f1 x l}
X0:{l : R | derivable_pt_abs f2 x l}
x0:R
p:derivable_pt_abs f1 x x0
x1:R
p0:derivable_pt_abs f2 x x1
{l : R | derivable_pt_abs (f1 - f2) x l}
  exists (x0 - x1).f1, f2:R -> R
x:R
X:{l : R | derivable_pt_abs f1 x l}
X0:{l : R | derivable_pt_abs f2 x l}
x0:R
p:derivable_pt_abs f1 x x0
x1:R
p0:derivable_pt_abs f2 x x1
derivable_pt_abs (f1 - f2) x (x0 - x1)
  apply derivable_pt_lim_minus; assumption.
Qed.

Lemma derivable_pt_mult :
  forall f1 f2 (x:R),
    derivable_pt f1 x -> derivable_pt f2 x -> derivable_pt (f1 * f2) x.forall (f1 f2 : R -> R) (x : R),
derivable_pt f1 x ->
derivable_pt f2 x -> derivable_pt (f1 * f2) x
Proof.forall (f1 f2 : R -> R) (x : R),
derivable_pt f1 x ->
derivable_pt f2 x -> derivable_pt (f1 * f2) x
  unfold derivable_pt; intros f1 f2 x X X0.f1, f2:R -> R
x:R
X:{l : R | derivable_pt_abs f1 x l}
X0:{l : R | derivable_pt_abs f2 x l}
{l : R | derivable_pt_abs (f1 * f2) x l}
  elim X; intros.f1, f2:R -> R
x:R
X:{l : R | derivable_pt_abs f1 x l}
X0:{l : R | derivable_pt_abs f2 x l}
x0:R
p:derivable_pt_abs f1 x x0
{l : R | derivable_pt_abs (f1 * f2) x l}
  elim X0; intros.f1, f2:R -> R
x:R
X:{l : R | derivable_pt_abs f1 x l}
X0:{l : R | derivable_pt_abs f2 x l}
x0:R
p:derivable_pt_abs f1 x x0
x1:R
p0:derivable_pt_abs f2 x x1
{l : R | derivable_pt_abs (f1 * f2) x l}
  exists (x0 * f2 x + f1 x * x1).f1, f2:R -> R
x:R
X:{l : R | derivable_pt_abs f1 x l}
X0:{l : R | derivable_pt_abs f2 x l}
x0:R
p:derivable_pt_abs f1 x x0
x1:R
p0:derivable_pt_abs f2 x x1
derivable_pt_abs (f1 * f2) x (x0 * f2 x + f1 x * x1)
  apply derivable_pt_lim_mult; assumption.
Qed.

Lemma derivable_pt_const : forall a x:R, derivable_pt (fct_cte a) x.forall a x : R, derivable_pt (fct_cte a) x
Proof.forall a x : R, derivable_pt (fct_cte a) x
  intros; unfold derivable_pt.a, x:R
{l : R | derivable_pt_abs (fct_cte a) x l}
  exists 0.a, x:R
derivable_pt_abs (fct_cte a) x 0
  apply derivable_pt_lim_const.
Qed.

Lemma derivable_pt_scal :
  forall f (a x:R), derivable_pt f x -> derivable_pt (mult_real_fct a f) x.forall (f : R -> R) (a x : R),
derivable_pt f x -> derivable_pt (mult_real_fct a f) x
Proof.forall (f : R -> R) (a x : R),
derivable_pt f x -> derivable_pt (mult_real_fct a f) x
  unfold derivable_pt; intros f1 a x X.f1:R -> R
a, x:R
X:{l : R | derivable_pt_abs f1 x l}
{l : R | derivable_pt_abs (mult_real_fct a f1) x l}
  elim X; intros.f1:R -> R
a, x:R
X:{l : R | derivable_pt_abs f1 x l}
x0:R
p:derivable_pt_abs f1 x x0
{l : R | derivable_pt_abs (mult_real_fct a f1) x l}
  exists (a * x0).f1:R -> R
a, x:R
X:{l : R | derivable_pt_abs f1 x l}
x0:R
p:derivable_pt_abs f1 x x0
derivable_pt_abs (mult_real_fct a f1) x (a * x0)
  apply derivable_pt_lim_scal; assumption.
Qed.

Lemma derivable_pt_id : forall x:R, derivable_pt id x.forall x : R, derivable_pt id x
Proof.forall x : R, derivable_pt id x
  unfold derivable_pt; intro.x:R
{l : R | derivable_pt_abs id x l}
  exists 1.x:R
derivable_pt_abs id x 1
  apply derivable_pt_lim_id.
Qed.

Lemma derivable_pt_Rsqr : forall x:R, derivable_pt Rsqr x.forall x : R, derivable_pt Rsqr x
Proof.forall x : R, derivable_pt Rsqr x
  unfold derivable_pt; intro; exists (2 * x).x:R
derivable_pt_abs Rsqr x (2 * x)
  apply derivable_pt_lim_Rsqr.
Qed.

Lemma derivable_pt_comp :
  forall f1 f2 (x:R),
    derivable_pt f1 x -> derivable_pt f2 (f1 x) -> derivable_pt (f2 o f1) x.forall (f1 f2 : R -> R) (x : R),
derivable_pt f1 x ->
derivable_pt f2 (f1 x) -> derivable_pt (f2 o f1) x
Proof.forall (f1 f2 : R -> R) (x : R),
derivable_pt f1 x ->
derivable_pt f2 (f1 x) -> derivable_pt (f2 o f1) x
  unfold derivable_pt; intros f1 f2 x X X0.f1, f2:R -> R
x:R
X:{l : R | derivable_pt_abs f1 x l}
X0:{l : R | derivable_pt_abs f2 (f1 x) l}
{l : R | derivable_pt_abs (f2 o f1) x l}
  elim X; intros.f1, f2:R -> R
x:R
X:{l : R | derivable_pt_abs f1 x l}
X0:{l : R | derivable_pt_abs f2 (f1 x) l}
x0:R
p:derivable_pt_abs f1 x x0
{l : R | derivable_pt_abs (f2 o f1) x l}
  elim X0; intros.f1, f2:R -> R
x:R
X:{l : R | derivable_pt_abs f1 x l}
X0:{l : R | derivable_pt_abs f2 (f1 x) l}
x0:R
p:derivable_pt_abs f1 x x0
x1:R
p0:derivable_pt_abs f2 (f1 x) x1
{l : R | derivable_pt_abs (f2 o f1) x l}
  exists (x1 * x0).f1, f2:R -> R
x:R
X:{l : R | derivable_pt_abs f1 x l}
X0:{l : R | derivable_pt_abs f2 (f1 x) l}
x0:R
p:derivable_pt_abs f1 x x0
x1:R
p0:derivable_pt_abs f2 (f1 x) x1
derivable_pt_abs (f2 o f1) x (x1 * x0)
  apply derivable_pt_lim_comp; assumption.
Qed.

Lemma derivable_plus :
  forall f1 f2, derivable f1 -> derivable f2 -> derivable (f1 + f2).forall f1 f2 : R -> R,
derivable f1 -> derivable f2 -> derivable (f1 + f2)
Proof.forall f1 f2 : R -> R,
derivable f1 -> derivable f2 -> derivable (f1 + f2)
  unfold derivable; intros f1 f2 X X0 x.f1, f2:R -> R
X:forall x0 : R, derivable_pt f1 x0
X0:forall x0 : R, derivable_pt f2 x0
x:R
derivable_pt (f1 + f2) x
  apply (derivable_pt_plus _ _ x (X _) (X0 _)).
Qed.

Lemma derivable_opp : forall f, derivable f -> derivable (- f).forall f : R -> R, derivable f -> derivable (- f)
Proof.forall f : R -> R, derivable f -> derivable (- f)
  unfold derivable; intros f X x.f:R -> R
X:forall x0 : R, derivable_pt f x0
x:R
derivable_pt (- f) x
  apply (derivable_pt_opp _ x (X _)).
Qed.

Lemma derivable_minus :
  forall f1 f2, derivable f1 -> derivable f2 -> derivable (f1 - f2).forall f1 f2 : R -> R,
derivable f1 -> derivable f2 -> derivable (f1 - f2)
Proof.forall f1 f2 : R -> R,
derivable f1 -> derivable f2 -> derivable (f1 - f2)
  unfold derivable; intros f1 f2 X X0 x.f1, f2:R -> R
X:forall x0 : R, derivable_pt f1 x0
X0:forall x0 : R, derivable_pt f2 x0
x:R
derivable_pt (f1 - f2) x
  apply (derivable_pt_minus _ _ x (X _) (X0 _)).
Qed.

Lemma derivable_mult :
  forall f1 f2, derivable f1 -> derivable f2 -> derivable (f1 * f2).forall f1 f2 : R -> R,
derivable f1 -> derivable f2 -> derivable (f1 * f2)
Proof.forall f1 f2 : R -> R,
derivable f1 -> derivable f2 -> derivable (f1 * f2)
  unfold derivable; intros f1 f2 X X0 x.f1, f2:R -> R
X:forall x0 : R, derivable_pt f1 x0
X0:forall x0 : R, derivable_pt f2 x0
x:R
derivable_pt (f1 * f2) x
  apply (derivable_pt_mult _ _ x (X _) (X0 _)).
Qed.

Lemma derivable_const : forall a:R, derivable (fct_cte a).forall a : R, derivable (fct_cte a)
Proof.forall a : R, derivable (fct_cte a)
  unfold derivable; intros.a, x:R
derivable_pt (fct_cte a) x
  apply derivable_pt_const.
Qed.

Lemma derivable_scal :
  forall f (a:R), derivable f -> derivable (mult_real_fct a f).forall (f : R -> R) (a : R),
derivable f -> derivable (mult_real_fct a f)
Proof.forall (f : R -> R) (a : R),
derivable f -> derivable (mult_real_fct a f)
  unfold derivable; intros f a X x.f:R -> R
a:R
X:forall x0 : R, derivable_pt f x0
x:R
derivable_pt (mult_real_fct a f) x
  apply (derivable_pt_scal _ a x (X _)).
Qed.

Lemma derivable_id : derivable id.derivable id
Proof.derivable id
  unfold derivable; intro; apply derivable_pt_id.
Qed.

Lemma derivable_Rsqr : derivable Rsqr.derivable Rsqr
Proof.derivable Rsqr
  unfold derivable; intro; apply derivable_pt_Rsqr.
Qed.

Lemma derivable_comp :
  forall f1 f2, derivable f1 -> derivable f2 -> derivable (f2 o f1).forall f1 f2 : R -> R,
derivable f1 -> derivable f2 -> derivable (f2 o f1)
Proof.forall f1 f2 : R -> R,
derivable f1 -> derivable f2 -> derivable (f2 o f1)
  unfold derivable; intros f1 f2 X X0 x.f1, f2:R -> R
X:forall x0 : R, derivable_pt f1 x0
X0:forall x0 : R, derivable_pt f2 x0
x:R
derivable_pt (f2 o f1) x
  apply (derivable_pt_comp _ _ x (X _) (X0 _)).
Qed.

Lemma derive_pt_plus :
  forall f1 f2 (x:R) (pr1:derivable_pt f1 x) (pr2:derivable_pt f2 x),
    derive_pt (f1 + f2) x (derivable_pt_plus _ _ _ pr1 pr2) =
    derive_pt f1 x pr1 + derive_pt f2 x pr2.forall (f1 f2 : R -> R) (x : R)
  (pr1 : derivable_pt f1 x) (pr2 : derivable_pt f2 x),
derive_pt (f1 + f2) x
  (derivable_pt_plus f1 f2 x pr1 pr2) =
derive_pt f1 x pr1 + derive_pt f2 x pr2
Proof.forall (f1 f2 : R -> R) (x : R)
  (pr1 : derivable_pt f1 x) (pr2 : derivable_pt f2 x),
derive_pt (f1 + f2) x
  (derivable_pt_plus f1 f2 x pr1 pr2) =
derive_pt f1 x pr1 + derive_pt f2 x pr2
  intros.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
derive_pt (f1 + f2) x
  (derivable_pt_plus f1 f2 x pr1 pr2) =
derive_pt f1 x pr1 + derive_pt f2 x pr2
  assert (H := derivable_derive f1 x pr1).f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
H:exists l : R, derive_pt f1 x pr1 = l
derive_pt (f1 + f2) x
  (derivable_pt_plus f1 f2 x pr1 pr2) =
derive_pt f1 x pr1 + derive_pt f2 x pr2
  assert (H0 := derivable_derive f2 x pr2).f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
H:exists l : R, derive_pt f1 x pr1 = l
H0:exists l : R, derive_pt f2 x pr2 = l
derive_pt (f1 + f2) x
  (derivable_pt_plus f1 f2 x pr1 pr2) =
derive_pt f1 x pr1 + derive_pt f2 x pr2
  assert
    (H1 := derivable_derive (f1 + f2)%F x (derivable_pt_plus _ _ _ pr1 pr2)).f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
H:exists l : R, derive_pt f1 x pr1 = l
H0:exists l : R, derive_pt f2 x pr2 = l
H1:exists l : R,
  derive_pt (f1 + f2) x
    (derivable_pt_plus f1 f2 x pr1 pr2) = l
derive_pt (f1 + f2) x
  (derivable_pt_plus f1 f2 x pr1 pr2) =
derive_pt f1 x pr1 + derive_pt f2 x pr2
  elim H; clear H; intros l1 H.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
H0:exists l : R, derive_pt f2 x pr2 = l
H1:exists l : R,
  derive_pt (f1 + f2) x
    (derivable_pt_plus f1 f2 x pr1 pr2) = l
l1:R
H:derive_pt f1 x pr1 = l1
derive_pt (f1 + f2) x
  (derivable_pt_plus f1 f2 x pr1 pr2) =
derive_pt f1 x pr1 + derive_pt f2 x pr2
  elim H0; clear H0; intros l2 H0.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
H1:exists l : R,
  derive_pt (f1 + f2) x
    (derivable_pt_plus f1 f2 x pr1 pr2) = l
l1:R
H:derive_pt f1 x pr1 = l1
l2:R
H0:derive_pt f2 x pr2 = l2
derive_pt (f1 + f2) x
  (derivable_pt_plus f1 f2 x pr1 pr2) =
derive_pt f1 x pr1 + derive_pt f2 x pr2
  elim H1; clear H1; intros l H1.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
l1:R
H:derive_pt f1 x pr1 = l1
l2:R
H0:derive_pt f2 x pr2 = l2
l:R
H1:derive_pt (f1 + f2) x
  (derivable_pt_plus f1 f2 x pr1 pr2) = l
derive_pt (f1 + f2) x
  (derivable_pt_plus f1 f2 x pr1 pr2) =
derive_pt f1 x pr1 + derive_pt f2 x pr2
  rewrite H; rewrite H0; apply derive_pt_eq_0.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
l1:R
H:derive_pt f1 x pr1 = l1
l2:R
H0:derive_pt f2 x pr2 = l2
l:R
H1:derive_pt (f1 + f2) x
  (derivable_pt_plus f1 f2 x pr1 pr2) = l
derivable_pt_lim (f1 + f2) x (l1 + l2)
  assert (H3 := proj2_sig pr1).f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
l1:R
H:derive_pt f1 x pr1 = l1
l2:R
H0:derive_pt f2 x pr2 = l2
l:R
H1:derive_pt (f1 + f2) x
  (derivable_pt_plus f1 f2 x pr1 pr2) = l
H3:(fun l0 : R => derivable_pt_abs f1 x l0)
  (proj1_sig pr1)
derivable_pt_lim (f1 + f2) x (l1 + l2)
  unfold derive_pt in H; rewrite H in H3.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
l1:R
H:proj1_sig pr1 = l1
l2:R
H0:derive_pt f2 x pr2 = l2
l:R
H1:derive_pt (f1 + f2) x
  (derivable_pt_plus f1 f2 x pr1 pr2) = l
H3:derivable_pt_abs f1 x l1
derivable_pt_lim (f1 + f2) x (l1 + l2)
  assert (H4 := proj2_sig pr2).f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
l1:R
H:proj1_sig pr1 = l1
l2:R
H0:derive_pt f2 x pr2 = l2
l:R
H1:derive_pt (f1 + f2) x
  (derivable_pt_plus f1 f2 x pr1 pr2) = l
H3:derivable_pt_abs f1 x l1
H4:(fun l0 : R => derivable_pt_abs f2 x l0)
  (proj1_sig pr2)
derivable_pt_lim (f1 + f2) x (l1 + l2)
  unfold derive_pt in H0; rewrite H0 in H4.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
l1:R
H:proj1_sig pr1 = l1
l2:R
H0:proj1_sig pr2 = l2
l:R
H1:derive_pt (f1 + f2) x
  (derivable_pt_plus f1 f2 x pr1 pr2) = l
H3:derivable_pt_abs f1 x l1
H4:derivable_pt_abs f2 x l2
derivable_pt_lim (f1 + f2) x (l1 + l2)
  apply derivable_pt_lim_plus; assumption.
Qed.

Lemma derive_pt_opp :
  forall f (x:R) (pr1:derivable_pt f x),
    derive_pt (- f) x (derivable_pt_opp _ _ pr1) = - derive_pt f x pr1.forall (f : R -> R) (x : R) (pr1 : derivable_pt f x),
derive_pt (- f) x (derivable_pt_opp f x pr1) =
- derive_pt f x pr1
Proof.forall (f : R -> R) (x : R) (pr1 : derivable_pt f x),
derive_pt (- f) x (derivable_pt_opp f x pr1) =
- derive_pt f x pr1
  intros.f:R -> R
x:R
pr1:derivable_pt f x
derive_pt (- f) x (derivable_pt_opp f x pr1) =
- derive_pt f x pr1
  assert (H := derivable_derive f x pr1).f:R -> R
x:R
pr1:derivable_pt f x
H:exists l : R, derive_pt f x pr1 = l
derive_pt (- f) x (derivable_pt_opp f x pr1) =
- derive_pt f x pr1
  assert (H0 := derivable_derive (- f)%F x (derivable_pt_opp _ _ pr1)).f:R -> R
x:R
pr1:derivable_pt f x
H:exists l : R, derive_pt f x pr1 = l
H0:exists l : R,
  derive_pt (- f) x (derivable_pt_opp f x pr1) =
  l
derive_pt (- f) x (derivable_pt_opp f x pr1) =
- derive_pt f x pr1
  elim H; clear H; intros l1 H.f:R -> R
x:R
pr1:derivable_pt f x
H0:exists l : R,
  derive_pt (- f) x (derivable_pt_opp f x pr1) =
  l
l1:R
H:derive_pt f x pr1 = l1
derive_pt (- f) x (derivable_pt_opp f x pr1) =
- derive_pt f x pr1
  elim H0; clear H0; intros l2 H0.f:R -> R
x:R
pr1:derivable_pt f x
l1:R
H:derive_pt f x pr1 = l1
l2:R
H0:derive_pt (- f) x (derivable_pt_opp f x pr1) = l2
derive_pt (- f) x (derivable_pt_opp f x pr1) =
- derive_pt f x pr1
  rewrite H; apply derive_pt_eq_0.f:R -> R
x:R
pr1:derivable_pt f x
l1:R
H:derive_pt f x pr1 = l1
l2:R
H0:derive_pt (- f) x (derivable_pt_opp f x pr1) = l2
derivable_pt_lim (- f) x (- l1)
  assert (H3 := proj2_sig pr1).f:R -> R
x:R
pr1:derivable_pt f x
l1:R
H:derive_pt f x pr1 = l1
l2:R
H0:derive_pt (- f) x (derivable_pt_opp f x pr1) = l2
H3:(fun l : R => derivable_pt_abs f x l)
  (proj1_sig pr1)
derivable_pt_lim (- f) x (- l1)
  unfold derive_pt in H; rewrite H in H3.f:R -> R
x:R
pr1:derivable_pt f x
l1:R
H:proj1_sig pr1 = l1
l2:R
H0:derive_pt (- f) x (derivable_pt_opp f x pr1) = l2
H3:derivable_pt_abs f x l1
derivable_pt_lim (- f) x (- l1)
  apply derivable_pt_lim_opp; assumption.
Qed.

Lemma derive_pt_minus :
  forall f1 f2 (x:R) (pr1:derivable_pt f1 x) (pr2:derivable_pt f2 x),
    derive_pt (f1 - f2) x (derivable_pt_minus _ _ _ pr1 pr2) =
    derive_pt f1 x pr1 - derive_pt f2 x pr2.forall (f1 f2 : R -> R) (x : R)
  (pr1 : derivable_pt f1 x) (pr2 : derivable_pt f2 x),
derive_pt (f1 - f2) x
  (derivable_pt_minus f1 f2 x pr1 pr2) =
derive_pt f1 x pr1 - derive_pt f2 x pr2
Proof.forall (f1 f2 : R -> R) (x : R)
  (pr1 : derivable_pt f1 x) (pr2 : derivable_pt f2 x),
derive_pt (f1 - f2) x
  (derivable_pt_minus f1 f2 x pr1 pr2) =
derive_pt f1 x pr1 - derive_pt f2 x pr2
  intros.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
derive_pt (f1 - f2) x
  (derivable_pt_minus f1 f2 x pr1 pr2) =
derive_pt f1 x pr1 - derive_pt f2 x pr2
  assert (H := derivable_derive f1 x pr1).f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
H:exists l : R, derive_pt f1 x pr1 = l
derive_pt (f1 - f2) x
  (derivable_pt_minus f1 f2 x pr1 pr2) =
derive_pt f1 x pr1 - derive_pt f2 x pr2
  assert (H0 := derivable_derive f2 x pr2).f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
H:exists l : R, derive_pt f1 x pr1 = l
H0:exists l : R, derive_pt f2 x pr2 = l
derive_pt (f1 - f2) x
  (derivable_pt_minus f1 f2 x pr1 pr2) =
derive_pt f1 x pr1 - derive_pt f2 x pr2
  assert
    (H1 := derivable_derive (f1 - f2)%F x (derivable_pt_minus _ _ _ pr1 pr2)).f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
H:exists l : R, derive_pt f1 x pr1 = l
H0:exists l : R, derive_pt f2 x pr2 = l
H1:exists l : R,
  derive_pt (f1 - f2) x
    (derivable_pt_minus f1 f2 x pr1 pr2) = l
derive_pt (f1 - f2) x
  (derivable_pt_minus f1 f2 x pr1 pr2) =
derive_pt f1 x pr1 - derive_pt f2 x pr2
  elim H; clear H; intros l1 H.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
H0:exists l : R, derive_pt f2 x pr2 = l
H1:exists l : R,
  derive_pt (f1 - f2) x
    (derivable_pt_minus f1 f2 x pr1 pr2) = l
l1:R
H:derive_pt f1 x pr1 = l1
derive_pt (f1 - f2) x
  (derivable_pt_minus f1 f2 x pr1 pr2) =
derive_pt f1 x pr1 - derive_pt f2 x pr2
  elim H0; clear H0; intros l2 H0.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
H1:exists l : R,
  derive_pt (f1 - f2) x
    (derivable_pt_minus f1 f2 x pr1 pr2) = l
l1:R
H:derive_pt f1 x pr1 = l1
l2:R
H0:derive_pt f2 x pr2 = l2
derive_pt (f1 - f2) x
  (derivable_pt_minus f1 f2 x pr1 pr2) =
derive_pt f1 x pr1 - derive_pt f2 x pr2
  elim H1; clear H1; intros l H1.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
l1:R
H:derive_pt f1 x pr1 = l1
l2:R
H0:derive_pt f2 x pr2 = l2
l:R
H1:derive_pt (f1 - f2) x
  (derivable_pt_minus f1 f2 x pr1 pr2) = l
derive_pt (f1 - f2) x
  (derivable_pt_minus f1 f2 x pr1 pr2) =
derive_pt f1 x pr1 - derive_pt f2 x pr2
  rewrite H; rewrite H0; apply derive_pt_eq_0.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
l1:R
H:derive_pt f1 x pr1 = l1
l2:R
H0:derive_pt f2 x pr2 = l2
l:R
H1:derive_pt (f1 - f2) x
  (derivable_pt_minus f1 f2 x pr1 pr2) = l
derivable_pt_lim (f1 - f2) x (l1 - l2)
  assert (H3 := proj2_sig pr1).f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
l1:R
H:derive_pt f1 x pr1 = l1
l2:R
H0:derive_pt f2 x pr2 = l2
l:R
H1:derive_pt (f1 - f2) x
  (derivable_pt_minus f1 f2 x pr1 pr2) = l
H3:(fun l0 : R => derivable_pt_abs f1 x l0)
  (proj1_sig pr1)
derivable_pt_lim (f1 - f2) x (l1 - l2)
  unfold derive_pt in H; rewrite H in H3.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
l1:R
H:proj1_sig pr1 = l1
l2:R
H0:derive_pt f2 x pr2 = l2
l:R
H1:derive_pt (f1 - f2) x
  (derivable_pt_minus f1 f2 x pr1 pr2) = l
H3:derivable_pt_abs f1 x l1
derivable_pt_lim (f1 - f2) x (l1 - l2)
  assert (H4 := proj2_sig pr2).f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
l1:R
H:proj1_sig pr1 = l1
l2:R
H0:derive_pt f2 x pr2 = l2
l:R
H1:derive_pt (f1 - f2) x
  (derivable_pt_minus f1 f2 x pr1 pr2) = l
H3:derivable_pt_abs f1 x l1
H4:(fun l0 : R => derivable_pt_abs f2 x l0)
  (proj1_sig pr2)
derivable_pt_lim (f1 - f2) x (l1 - l2)
  unfold derive_pt in H0; rewrite H0 in H4.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
l1:R
H:proj1_sig pr1 = l1
l2:R
H0:proj1_sig pr2 = l2
l:R
H1:derive_pt (f1 - f2) x
  (derivable_pt_minus f1 f2 x pr1 pr2) = l
H3:derivable_pt_abs f1 x l1
H4:derivable_pt_abs f2 x l2
derivable_pt_lim (f1 - f2) x (l1 - l2)
  apply derivable_pt_lim_minus; assumption.
Qed.

Lemma derive_pt_mult :
  forall f1 f2 (x:R) (pr1:derivable_pt f1 x) (pr2:derivable_pt f2 x),
    derive_pt (f1 * f2) x (derivable_pt_mult _ _ _ pr1 pr2) =
    derive_pt f1 x pr1 * f2 x + f1 x * derive_pt f2 x pr2.forall (f1 f2 : R -> R) (x : R)
  (pr1 : derivable_pt f1 x) (pr2 : derivable_pt f2 x),
derive_pt (f1 * f2) x
  (derivable_pt_mult f1 f2 x pr1 pr2) =
derive_pt f1 x pr1 * f2 x + f1 x * derive_pt f2 x pr2
Proof.forall (f1 f2 : R -> R) (x : R)
  (pr1 : derivable_pt f1 x) (pr2 : derivable_pt f2 x),
derive_pt (f1 * f2) x
  (derivable_pt_mult f1 f2 x pr1 pr2) =
derive_pt f1 x pr1 * f2 x + f1 x * derive_pt f2 x pr2
  intros.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
derive_pt (f1 * f2) x
  (derivable_pt_mult f1 f2 x pr1 pr2) =
derive_pt f1 x pr1 * f2 x + f1 x * derive_pt f2 x pr2
  assert (H := derivable_derive f1 x pr1).f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
H:exists l : R, derive_pt f1 x pr1 = l
derive_pt (f1 * f2) x
  (derivable_pt_mult f1 f2 x pr1 pr2) =
derive_pt f1 x pr1 * f2 x + f1 x * derive_pt f2 x pr2
  assert (H0 := derivable_derive f2 x pr2).f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
H:exists l : R, derive_pt f1 x pr1 = l
H0:exists l : R, derive_pt f2 x pr2 = l
derive_pt (f1 * f2) x
  (derivable_pt_mult f1 f2 x pr1 pr2) =
derive_pt f1 x pr1 * f2 x + f1 x * derive_pt f2 x pr2
  assert
    (H1 := derivable_derive (f1 * f2)%F x (derivable_pt_mult _ _ _ pr1 pr2)).f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
H:exists l : R, derive_pt f1 x pr1 = l
H0:exists l : R, derive_pt f2 x pr2 = l
H1:exists l : R,
  derive_pt (f1 * f2) x
    (derivable_pt_mult f1 f2 x pr1 pr2) = l
derive_pt (f1 * f2) x
  (derivable_pt_mult f1 f2 x pr1 pr2) =
derive_pt f1 x pr1 * f2 x + f1 x * derive_pt f2 x pr2
  elim H; clear H; intros l1 H.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
H0:exists l : R, derive_pt f2 x pr2 = l
H1:exists l : R,
  derive_pt (f1 * f2) x
    (derivable_pt_mult f1 f2 x pr1 pr2) = l
l1:R
H:derive_pt f1 x pr1 = l1
derive_pt (f1 * f2) x
  (derivable_pt_mult f1 f2 x pr1 pr2) =
derive_pt f1 x pr1 * f2 x + f1 x * derive_pt f2 x pr2
  elim H0; clear H0; intros l2 H0.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
H1:exists l : R,
  derive_pt (f1 * f2) x
    (derivable_pt_mult f1 f2 x pr1 pr2) = l
l1:R
H:derive_pt f1 x pr1 = l1
l2:R
H0:derive_pt f2 x pr2 = l2
derive_pt (f1 * f2) x
  (derivable_pt_mult f1 f2 x pr1 pr2) =
derive_pt f1 x pr1 * f2 x + f1 x * derive_pt f2 x pr2
  elim H1; clear H1; intros l H1.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
l1:R
H:derive_pt f1 x pr1 = l1
l2:R
H0:derive_pt f2 x pr2 = l2
l:R
H1:derive_pt (f1 * f2) x
  (derivable_pt_mult f1 f2 x pr1 pr2) = l
derive_pt (f1 * f2) x
  (derivable_pt_mult f1 f2 x pr1 pr2) =
derive_pt f1 x pr1 * f2 x + f1 x * derive_pt f2 x pr2
  rewrite H; rewrite H0; apply derive_pt_eq_0.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
l1:R
H:derive_pt f1 x pr1 = l1
l2:R
H0:derive_pt f2 x pr2 = l2
l:R
H1:derive_pt (f1 * f2) x
  (derivable_pt_mult f1 f2 x pr1 pr2) = l
derivable_pt_lim (f1 * f2) x (l1 * f2 x + f1 x * l2)
  assert (H3 := proj2_sig pr1).f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
l1:R
H:derive_pt f1 x pr1 = l1
l2:R
H0:derive_pt f2 x pr2 = l2
l:R
H1:derive_pt (f1 * f2) x
  (derivable_pt_mult f1 f2 x pr1 pr2) = l
H3:(fun l0 : R => derivable_pt_abs f1 x l0)
  (proj1_sig pr1)
derivable_pt_lim (f1 * f2) x (l1 * f2 x + f1 x * l2)
  unfold derive_pt in H; rewrite H in H3.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
l1:R
H:proj1_sig pr1 = l1
l2:R
H0:derive_pt f2 x pr2 = l2
l:R
H1:derive_pt (f1 * f2) x
  (derivable_pt_mult f1 f2 x pr1 pr2) = l
H3:derivable_pt_abs f1 x l1
derivable_pt_lim (f1 * f2) x (l1 * f2 x + f1 x * l2)
  assert (H4 := proj2_sig pr2).f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
l1:R
H:proj1_sig pr1 = l1
l2:R
H0:derive_pt f2 x pr2 = l2
l:R
H1:derive_pt (f1 * f2) x
  (derivable_pt_mult f1 f2 x pr1 pr2) = l
H3:derivable_pt_abs f1 x l1
H4:(fun l0 : R => derivable_pt_abs f2 x l0)
  (proj1_sig pr2)
derivable_pt_lim (f1 * f2) x (l1 * f2 x + f1 x * l2)
  unfold derive_pt in H0; rewrite H0 in H4.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 x
l1:R
H:proj1_sig pr1 = l1
l2:R
H0:proj1_sig pr2 = l2
l:R
H1:derive_pt (f1 * f2) x
  (derivable_pt_mult f1 f2 x pr1 pr2) = l
H3:derivable_pt_abs f1 x l1
H4:derivable_pt_abs f2 x l2
derivable_pt_lim (f1 * f2) x (l1 * f2 x + f1 x * l2)
  apply derivable_pt_lim_mult; assumption.
Qed.

Lemma derive_pt_const :
  forall a x:R, derive_pt (fct_cte a) x (derivable_pt_const a x) = 0.forall a x : R,
derive_pt (fct_cte a) x (derivable_pt_const a x) = 0
Proof.forall a x : R,
derive_pt (fct_cte a) x (derivable_pt_const a x) = 0
  intros.a, x:R
derive_pt (fct_cte a) x (derivable_pt_const a x) = 0
  apply derive_pt_eq_0.a, x:R
derivable_pt_lim (fct_cte a) x 0
  apply derivable_pt_lim_const.
Qed.

Lemma derive_pt_scal :
  forall f (a x:R) (pr:derivable_pt f x),
    derive_pt (mult_real_fct a f) x (derivable_pt_scal _ _ _ pr) =
    a * derive_pt f x pr.forall (f : R -> R) (a x : R) (pr : derivable_pt f x),
derive_pt (mult_real_fct a f) x
  (derivable_pt_scal f a x pr) = a * derive_pt f x pr
Proof.forall (f : R -> R) (a x : R) (pr : derivable_pt f x),
derive_pt (mult_real_fct a f) x
  (derivable_pt_scal f a x pr) = a * derive_pt f x pr
  intros.f:R -> R
a, x:R
pr:derivable_pt f x
derive_pt (mult_real_fct a f) x
  (derivable_pt_scal f a x pr) = a * derive_pt f x pr
  assert (H := derivable_derive f x pr).f:R -> R
a, x:R
pr:derivable_pt f x
H:exists l : R, derive_pt f x pr = l
derive_pt (mult_real_fct a f) x
  (derivable_pt_scal f a x pr) = a * derive_pt f x pr
  assert
    (H0 := derivable_derive (mult_real_fct a f) x (derivable_pt_scal _ _ _ pr)).f:R -> R
a, x:R
pr:derivable_pt f x
H:exists l : R, derive_pt f x pr = l
H0:exists l : R,
  derive_pt (mult_real_fct a f) x
    (derivable_pt_scal f a x pr) = l
derive_pt (mult_real_fct a f) x
  (derivable_pt_scal f a x pr) = a * derive_pt f x pr
  elim H; clear H; intros l1 H.f:R -> R
a, x:R
pr:derivable_pt f x
H0:exists l : R,
  derive_pt (mult_real_fct a f) x
    (derivable_pt_scal f a x pr) = l
l1:R
H:derive_pt f x pr = l1
derive_pt (mult_real_fct a f) x
  (derivable_pt_scal f a x pr) = a * derive_pt f x pr
  elim H0; clear H0; intros l2 H0.f:R -> R
a, x:R
pr:derivable_pt f x
l1:R
H:derive_pt f x pr = l1
l2:R
H0:derive_pt (mult_real_fct a f) x
  (derivable_pt_scal f a x pr) = l2
derive_pt (mult_real_fct a f) x
  (derivable_pt_scal f a x pr) = a * derive_pt f x pr
  rewrite H; apply derive_pt_eq_0.f:R -> R
a, x:R
pr:derivable_pt f x
l1:R
H:derive_pt f x pr = l1
l2:R
H0:derive_pt (mult_real_fct a f) x
  (derivable_pt_scal f a x pr) = l2
derivable_pt_lim (mult_real_fct a f) x (a * l1)
  assert (H3 := proj2_sig pr).f:R -> R
a, x:R
pr:derivable_pt f x
l1:R
H:derive_pt f x pr = l1
l2:R
H0:derive_pt (mult_real_fct a f) x
  (derivable_pt_scal f a x pr) = l2
H3:(fun l : R => derivable_pt_abs f x l)
  (proj1_sig pr)
derivable_pt_lim (mult_real_fct a f) x (a * l1)
  unfold derive_pt in H; rewrite H in H3.f:R -> R
a, x:R
pr:derivable_pt f x
l1:R
H:proj1_sig pr = l1
l2:R
H0:derive_pt (mult_real_fct a f) x
  (derivable_pt_scal f a x pr) = l2
H3:derivable_pt_abs f x l1
derivable_pt_lim (mult_real_fct a f) x (a * l1)
  apply derivable_pt_lim_scal; assumption.
Qed.

Lemma derive_pt_id : forall x:R, derive_pt id x (derivable_pt_id _) = 1.forall x : R, derive_pt id x (derivable_pt_id x) = 1
Proof.forall x : R, derive_pt id x (derivable_pt_id x) = 1
  intros.x:R
derive_pt id x (derivable_pt_id x) = 1
  apply derive_pt_eq_0.x:R
derivable_pt_lim id x 1
  apply derivable_pt_lim_id.
Qed.

Lemma derive_pt_Rsqr :
  forall x:R, derive_pt Rsqr x (derivable_pt_Rsqr _) = 2 * x.forall x : R,
derive_pt Rsqr x (derivable_pt_Rsqr x) = 2 * x
Proof.forall x : R,
derive_pt Rsqr x (derivable_pt_Rsqr x) = 2 * x
  intros.x:R
derive_pt Rsqr x (derivable_pt_Rsqr x) = 2 * x
  apply derive_pt_eq_0.x:R
derivable_pt_lim Rsqr x (2 * x)
  apply derivable_pt_lim_Rsqr.
Qed.

Lemma derive_pt_comp :
  forall f1 f2 (x:R) (pr1:derivable_pt f1 x) (pr2:derivable_pt f2 (f1 x)),
    derive_pt (f2 o f1) x (derivable_pt_comp _ _ _ pr1 pr2) =
    derive_pt f2 (f1 x) pr2 * derive_pt f1 x pr1.forall (f1 f2 : R -> R) (x : R)
  (pr1 : derivable_pt f1 x)
  (pr2 : derivable_pt f2 (f1 x)),
derive_pt (f2 o f1) x
  (derivable_pt_comp f1 f2 x pr1 pr2) =
derive_pt f2 (f1 x) pr2 * derive_pt f1 x pr1
Proof.forall (f1 f2 : R -> R) (x : R)
  (pr1 : derivable_pt f1 x)
  (pr2 : derivable_pt f2 (f1 x)),
derive_pt (f2 o f1) x
  (derivable_pt_comp f1 f2 x pr1 pr2) =
derive_pt f2 (f1 x) pr2 * derive_pt f1 x pr1
  intros.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 (f1 x)
derive_pt (f2 o f1) x
  (derivable_pt_comp f1 f2 x pr1 pr2) =
derive_pt f2 (f1 x) pr2 * derive_pt f1 x pr1
  assert (H := derivable_derive f1 x pr1).f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 (f1 x)
H:exists l : R, derive_pt f1 x pr1 = l
derive_pt (f2 o f1) x
  (derivable_pt_comp f1 f2 x pr1 pr2) =
derive_pt f2 (f1 x) pr2 * derive_pt f1 x pr1
  assert (H0 := derivable_derive f2 (f1 x) pr2).f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 (f1 x)
H:exists l : R, derive_pt f1 x pr1 = l
H0:exists l : R, derive_pt f2 (f1 x) pr2 = l
derive_pt (f2 o f1) x
  (derivable_pt_comp f1 f2 x pr1 pr2) =
derive_pt f2 (f1 x) pr2 * derive_pt f1 x pr1
  assert
    (H1 := derivable_derive (f2 o f1)%F x (derivable_pt_comp _ _ _ pr1 pr2)).f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 (f1 x)
H:exists l : R, derive_pt f1 x pr1 = l
H0:exists l : R, derive_pt f2 (f1 x) pr2 = l
H1:exists l : R,
  derive_pt (f2 o f1) x
    (derivable_pt_comp f1 f2 x pr1 pr2) = l
derive_pt (f2 o f1) x
  (derivable_pt_comp f1 f2 x pr1 pr2) =
derive_pt f2 (f1 x) pr2 * derive_pt f1 x pr1
  elim H; clear H; intros l1 H.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 (f1 x)
H0:exists l : R, derive_pt f2 (f1 x) pr2 = l
H1:exists l : R,
  derive_pt (f2 o f1) x
    (derivable_pt_comp f1 f2 x pr1 pr2) = l
l1:R
H:derive_pt f1 x pr1 = l1
derive_pt (f2 o f1) x
  (derivable_pt_comp f1 f2 x pr1 pr2) =
derive_pt f2 (f1 x) pr2 * derive_pt f1 x pr1
  elim H0; clear H0; intros l2 H0.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 (f1 x)
H1:exists l : R,
  derive_pt (f2 o f1) x
    (derivable_pt_comp f1 f2 x pr1 pr2) = l
l1:R
H:derive_pt f1 x pr1 = l1
l2:R
H0:derive_pt f2 (f1 x) pr2 = l2
derive_pt (f2 o f1) x
  (derivable_pt_comp f1 f2 x pr1 pr2) =
derive_pt f2 (f1 x) pr2 * derive_pt f1 x pr1
  elim H1; clear H1; intros l H1.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 (f1 x)
l1:R
H:derive_pt f1 x pr1 = l1
l2:R
H0:derive_pt f2 (f1 x) pr2 = l2
l:R
H1:derive_pt (f2 o f1) x
  (derivable_pt_comp f1 f2 x pr1 pr2) = l
derive_pt (f2 o f1) x
  (derivable_pt_comp f1 f2 x pr1 pr2) =
derive_pt f2 (f1 x) pr2 * derive_pt f1 x pr1
  rewrite H; rewrite H0; apply derive_pt_eq_0.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 (f1 x)
l1:R
H:derive_pt f1 x pr1 = l1
l2:R
H0:derive_pt f2 (f1 x) pr2 = l2
l:R
H1:derive_pt (f2 o f1) x
  (derivable_pt_comp f1 f2 x pr1 pr2) = l
derivable_pt_lim (f2 o f1) x (l2 * l1)
  assert (H3 := proj2_sig pr1).f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 (f1 x)
l1:R
H:derive_pt f1 x pr1 = l1
l2:R
H0:derive_pt f2 (f1 x) pr2 = l2
l:R
H1:derive_pt (f2 o f1) x
  (derivable_pt_comp f1 f2 x pr1 pr2) = l
H3:(fun l0 : R => derivable_pt_abs f1 x l0)
  (proj1_sig pr1)
derivable_pt_lim (f2 o f1) x (l2 * l1)
  unfold derive_pt in H; rewrite H in H3.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 (f1 x)
l1:R
H:proj1_sig pr1 = l1
l2:R
H0:derive_pt f2 (f1 x) pr2 = l2
l:R
H1:derive_pt (f2 o f1) x
  (derivable_pt_comp f1 f2 x pr1 pr2) = l
H3:derivable_pt_abs f1 x l1
derivable_pt_lim (f2 o f1) x (l2 * l1)
  assert (H4 := proj2_sig pr2).f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 (f1 x)
l1:R
H:proj1_sig pr1 = l1
l2:R
H0:derive_pt f2 (f1 x) pr2 = l2
l:R
H1:derive_pt (f2 o f1) x
  (derivable_pt_comp f1 f2 x pr1 pr2) = l
H3:derivable_pt_abs f1 x l1
H4:(fun l0 : R => derivable_pt_abs f2 (f1 x) l0)
  (proj1_sig pr2)
derivable_pt_lim (f2 o f1) x (l2 * l1)
  unfold derive_pt in H0; rewrite H0 in H4.f1, f2:R -> R
x:R
pr1:derivable_pt f1 x
pr2:derivable_pt f2 (f1 x)
l1:R
H:proj1_sig pr1 = l1
l2:R
H0:proj1_sig pr2 = l2
l:R
H1:derive_pt (f2 o f1) x
  (derivable_pt_comp f1 f2 x pr1 pr2) = l
H3:derivable_pt_abs f1 x l1
H4:derivable_pt_abs f2 (f1 x) l2
derivable_pt_lim (f2 o f1) x (l2 * l1)
  apply derivable_pt_lim_comp; assumption.
Qed.

(* Pow *)
Definition pow_fct (n:nat) (y:R) : R := y ^ n.

Lemma derivable_pt_lim_pow_pos :
  forall (x:R) (n:nat),
    (0 < n)%nat -> derivable_pt_lim (fun y:R => y ^ n) x (INR n * x ^ pred n).forall (x : R) (n : nat),
(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
Proof.forall (x : R) (n : nat),
(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
  intros.x:R
n:nat
H:(0 < n)%nat
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
  induction  n as [| n Hrecn].x:R
H:(0 < 0)%nat
derivable_pt_lim (fun y : R => y ^ 0) x
  (INR 0 * x ^ Init.Nat.pred 0)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
derivable_pt_lim (fun y : R => y ^ S n) x
  (INR (S n) * x ^ Init.Nat.pred (S n))
  elim (lt_irrefl _ H).x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
derivable_pt_lim (fun y : R => y ^ S n) x
  (INR (S n) * x ^ Init.Nat.pred (S n))
  cut (n = 0%nat \/ (0 < n)%nat).x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
n = 0%nat \/ (0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ S n) x
  (INR (S n) * x ^ Init.Nat.pred (S n))
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
n = 0%nat \/ (0 < n)%nat
  intro; elim H0; intro.x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:n = 0%nat
derivable_pt_lim (fun y : R => y ^ S n) x
  (INR (S n) * x ^ Init.Nat.pred (S n))
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
derivable_pt_lim (fun y : R => y ^ S n) x
  (INR (S n) * x ^ Init.Nat.pred (S n))
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
n = 0%nat \/ (0 < n)%nat
  rewrite H1; simpl.x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:n = 0%nat
derivable_pt_lim (fun y : R => y * 1) x (1 * 1)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
derivable_pt_lim (fun y : R => y ^ S n) x
  (INR (S n) * x ^ Init.Nat.pred (S n))
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
n = 0%nat \/ (0 < n)%nat
  replace (fun y:R => y * 1) with (id * fct_cte 1)%F.x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:n = 0%nat
derivable_pt_lim (id * fct_cte 1) x (1 * 1)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:n = 0%nat
(id * fct_cte 1)%F = (fun y : R => y * 1)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
derivable_pt_lim (fun y : R => y ^ S n) x
  (INR (S n) * x ^ Init.Nat.pred (S n))
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
n = 0%nat \/ (0 < n)%nat
  replace (1 * 1) with (1 * fct_cte 1 x + id x * 0).x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:n = 0%nat
derivable_pt_lim (id * fct_cte 1) x
  (1 * fct_cte 1 x + id x * 0)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:n = 0%nat
1 * fct_cte 1 x + id x * 0 = 1 * 1
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:n = 0%nat
(id * fct_cte 1)%F = (fun y : R => y * 1)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
derivable_pt_lim (fun y : R => y ^ S n) x
  (INR (S n) * x ^ Init.Nat.pred (S n))
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
n = 0%nat \/ (0 < n)%nat
  apply derivable_pt_lim_mult.x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:n = 0%nat
derivable_pt_lim id x 1
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:n = 0%nat
derivable_pt_lim (fct_cte 1) x 0
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:n = 0%nat
1 * fct_cte 1 x + id x * 0 = 1 * 1
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:n = 0%nat
(id * fct_cte 1)%F = (fun y : R => y * 1)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
derivable_pt_lim (fun y : R => y ^ S n) x
  (INR (S n) * x ^ Init.Nat.pred (S n))
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
n = 0%nat \/ (0 < n)%nat
  apply derivable_pt_lim_id.x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:n = 0%nat
derivable_pt_lim (fct_cte 1) x 0
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:n = 0%nat
1 * fct_cte 1 x + id x * 0 = 1 * 1
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:n = 0%nat
(id * fct_cte 1)%F = (fun y : R => y * 1)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
derivable_pt_lim (fun y : R => y ^ S n) x
  (INR (S n) * x ^ Init.Nat.pred (S n))
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
n = 0%nat \/ (0 < n)%nat
  apply derivable_pt_lim_const.x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:n = 0%nat
1 * fct_cte 1 x + id x * 0 = 1 * 1
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:n = 0%nat
(id * fct_cte 1)%F = (fun y : R => y * 1)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
derivable_pt_lim (fun y : R => y ^ S n) x
  (INR (S n) * x ^ Init.Nat.pred (S n))
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
n = 0%nat \/ (0 < n)%nat
  unfold fct_cte, id; ring.x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:n = 0%nat
(id * fct_cte 1)%F = (fun y : R => y * 1)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
derivable_pt_lim (fun y : R => y ^ S n) x
  (INR (S n) * x ^ Init.Nat.pred (S n))
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
n = 0%nat \/ (0 < n)%nat
  reflexivity.x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
derivable_pt_lim (fun y : R => y ^ S n) x
  (INR (S n) * x ^ Init.Nat.pred (S n))
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
n = 0%nat \/ (0 < n)%nat
  replace (fun y:R => y ^ S n) with (fun y:R => y * y ^ n).x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
derivable_pt_lim (fun y : R => y * y ^ n) x
  (INR (S n) * x ^ Init.Nat.pred (S n))
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
(fun y : R => y * y ^ n) = (fun y : R => y ^ S n)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
n = 0%nat \/ (0 < n)%nat
  replace (pred (S n)) with n; [ idtac | reflexivity ].x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
derivable_pt_lim (fun y : R => y * y ^ n) x
  (INR (S n) * x ^ n)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
(fun y : R => y * y ^ n) = (fun y : R => y ^ S n)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
n = 0%nat \/ (0 < n)%nat
  replace (fun y:R => y * y ^ n) with (id * (fun y:R => y ^ n))%F.x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
derivable_pt_lim (id * (fun y : R => y ^ n)) x
  (INR (S n) * x ^ n)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
(id * (fun y : R => y ^ n))%F =
(fun y : R => y * y ^ n)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
(fun y : R => y * y ^ n) = (fun y : R => y ^ S n)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
n = 0%nat \/ (0 < n)%nat
  set (f := fun y:R => y ^ n).x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
f:=fun y : R => y ^ n:R -> R
derivable_pt_lim (id * f) x (INR (S n) * x ^ n)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
(id * (fun y : R => y ^ n))%F =
(fun y : R => y * y ^ n)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
(fun y : R => y * y ^ n) = (fun y : R => y ^ S n)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
n = 0%nat \/ (0 < n)%nat
  replace (INR (S n) * x ^ n) with (1 * f x + id x * (INR n * x ^ pred n)).x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
f:=fun y : R => y ^ n:R -> R
derivable_pt_lim (id * f) x
  (1 * f x + id x * (INR n * x ^ Init.Nat.pred n))
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
f:=fun y : R => y ^ n:R -> R
1 * f x + id x * (INR n * x ^ Init.Nat.pred n) =
INR (S n) * x ^ n
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
(id * (fun y : R => y ^ n))%F =
(fun y : R => y * y ^ n)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
(fun y : R => y * y ^ n) = (fun y : R => y ^ S n)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
n = 0%nat \/ (0 < n)%nat
  apply derivable_pt_lim_mult.x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
f:=fun y : R => y ^ n:R -> R
derivable_pt_lim id x 1
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
f:=fun y : R => y ^ n:R -> R
derivable_pt_lim f x (INR n * x ^ Init.Nat.pred n)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
f:=fun y : R => y ^ n:R -> R
1 * f x + id x * (INR n * x ^ Init.Nat.pred n) =
INR (S n) * x ^ n
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
(id * (fun y : R => y ^ n))%F =
(fun y : R => y * y ^ n)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
(fun y : R => y * y ^ n) = (fun y : R => y ^ S n)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
n = 0%nat \/ (0 < n)%nat
  apply derivable_pt_lim_id.x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
f:=fun y : R => y ^ n:R -> R
derivable_pt_lim f x (INR n * x ^ Init.Nat.pred n)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
f:=fun y : R => y ^ n:R -> R
1 * f x + id x * (INR n * x ^ Init.Nat.pred n) =
INR (S n) * x ^ n
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
(id * (fun y : R => y ^ n))%F =
(fun y : R => y * y ^ n)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
(fun y : R => y * y ^ n) = (fun y : R => y ^ S n)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
n = 0%nat \/ (0 < n)%nat
  unfold f; apply Hrecn; assumption.x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
f:=fun y : R => y ^ n:R -> R
1 * f x + id x * (INR n * x ^ Init.Nat.pred n) =
INR (S n) * x ^ n
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
(id * (fun y : R => y ^ n))%F =
(fun y : R => y * y ^ n)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
(fun y : R => y * y ^ n) = (fun y : R => y ^ S n)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
n = 0%nat \/ (0 < n)%nat
  unfold f.x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
f:=fun y : R => y ^ n:R -> R
1 * x ^ n + id x * (INR n * x ^ Init.Nat.pred n) =
INR (S n) * x ^ n
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
(id * (fun y : R => y ^ n))%F =
(fun y : R => y * y ^ n)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
(fun y : R => y * y ^ n) = (fun y : R => y ^ S n)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
n = 0%nat \/ (0 < n)%nat
  pattern n at 1 5; replace n with (S (pred n)).x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
f:=fun y : R => y ^ n:R -> R
1 * x ^ S (Init.Nat.pred n) +
id x * (INR n * x ^ Init.Nat.pred n) =
INR (S n) * x ^ S (Init.Nat.pred n)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
f:=fun y : R => y ^ n:R -> R
S (Init.Nat.pred n) = n
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
(id * (fun y : R => y ^ n))%F =
(fun y : R => y * y ^ n)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
(fun y : R => y * y ^ n) = (fun y : R => y ^ S n)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
n = 0%nat \/ (0 < n)%nat
  unfold id; rewrite S_INR; simpl.x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
f:=fun y : R => y ^ n:R -> R
1 * (x * x ^ Init.Nat.pred n) +
x * (INR n * x ^ Init.Nat.pred n) =
(INR n + 1) * (x * x ^ Init.Nat.pred n)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
f:=fun y : R => y ^ n:R -> R
S (Init.Nat.pred n) = n
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
(id * (fun y : R => y ^ n))%F =
(fun y : R => y * y ^ n)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
(fun y : R => y * y ^ n) = (fun y : R => y ^ S n)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
n = 0%nat \/ (0 < n)%nat
  ring.x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
f:=fun y : R => y ^ n:R -> R
S (Init.Nat.pred n) = n
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
(id * (fun y : R => y ^ n))%F =
(fun y : R => y * y ^ n)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
(fun y : R => y * y ^ n) = (fun y : R => y ^ S n)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
n = 0%nat \/ (0 < n)%nat
  symmetry ; apply S_pred with 0%nat; assumption.x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
(id * (fun y : R => y ^ n))%F =
(fun y : R => y * y ^ n)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
(fun y : R => y * y ^ n) = (fun y : R => y ^ S n)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
n = 0%nat \/ (0 < n)%nat
  unfold mult_fct, id; reflexivity.x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H0:n = 0%nat \/ (0 < n)%nat
H1:(0 < n)%nat
(fun y : R => y * y ^ n) = (fun y : R => y ^ S n)
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
n = 0%nat \/ (0 < n)%nat
  reflexivity.x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
n = 0%nat \/ (0 < n)%nat
  inversion H.x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
H1:0%nat = n
0%nat = 0%nat \/ (0 < 0)%nat
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
m:nat
H1:(1 <= n)%nat
H0:m = n
n = 0%nat \/ (0 < n)%nat
  left; reflexivity.x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
m:nat
H1:(1 <= n)%nat
H0:m = n
n = 0%nat \/ (0 < n)%nat
  right.x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
m:nat
H1:(1 <= n)%nat
H0:m = n
(0 < n)%nat
  apply lt_le_trans with 1%nat.x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
m:nat
H1:(1 <= n)%nat
H0:m = n
(0 < 1)%nat
x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
m:nat
H1:(1 <= n)%nat
H0:m = n
(1 <= n)%nat
  apply lt_O_Sn.x:R
n:nat
H:(0 < S n)%nat
Hrecn:(0 < n)%nat ->
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
m:nat
H1:(1 <= n)%nat
H0:m = n
(1 <= n)%nat
  assumption.
Qed.

Lemma derivable_pt_lim_pow :
  forall (x:R) (n:nat),
    derivable_pt_lim (fun y:R => y ^ n) x (INR n * x ^ pred n).forall (x : R) (n : nat),
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
Proof.forall (x : R) (n : nat),
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
  intros.x:R
n:nat
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
  induction  n as [| n Hrecn].x:R
derivable_pt_lim (fun y : R => y ^ 0) x
  (INR 0 * x ^ Init.Nat.pred 0)
x:R
n:nat
Hrecn:derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
derivable_pt_lim (fun y : R => y ^ S n) x
  (INR (S n) * x ^ Init.Nat.pred (S n))
  simpl.x:R
derivable_pt_lim (fun _ : R => 1) x (0 * 1)
x:R
n:nat
Hrecn:derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
derivable_pt_lim (fun y : R => y ^ S n) x
  (INR (S n) * x ^ Init.Nat.pred (S n))
  rewrite Rmult_0_l.x:R
derivable_pt_lim (fun _ : R => 1) x 0
x:R
n:nat
Hrecn:derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
derivable_pt_lim (fun y : R => y ^ S n) x
  (INR (S n) * x ^ Init.Nat.pred (S n))
  replace (fun _:R => 1) with (fct_cte 1);
  [ apply derivable_pt_lim_const | reflexivity ].x:R
n:nat
Hrecn:derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
derivable_pt_lim (fun y : R => y ^ S n) x
  (INR (S n) * x ^ Init.Nat.pred (S n))
  apply derivable_pt_lim_pow_pos.x:R
n:nat
Hrecn:derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
(0 < S n)%nat
  apply lt_O_Sn.
Qed.

Lemma derivable_pt_pow :
  forall (n:nat) (x:R), derivable_pt (fun y:R => y ^ n) x.forall (n : nat) (x : R),
derivable_pt (fun y : R => y ^ n) x
Proof.forall (n : nat) (x : R),
derivable_pt (fun y : R => y ^ n) x
  intros; unfold derivable_pt.n:nat
x:R
{l : R | derivable_pt_abs (fun y : R => y ^ n) x l}
  exists (INR n * x ^ pred n).n:nat
x:R
derivable_pt_abs (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
  apply derivable_pt_lim_pow.
Qed.

Lemma derivable_pow : forall n:nat, derivable (fun y:R => y ^ n).forall n : nat, derivable (fun y : R => y ^ n)
Proof.forall n : nat, derivable (fun y : R => y ^ n)
  intro; unfold derivable; intro; apply derivable_pt_pow.
Qed.

Lemma derive_pt_pow :
  forall (n:nat) (x:R),
    derive_pt (fun y:R => y ^ n) x (derivable_pt_pow n x) = INR n * x ^ pred n.forall (n : nat) (x : R),
derive_pt (fun y : R => y ^ n) x
  (derivable_pt_pow n x) = INR n * x ^ Init.Nat.pred n
Proof.forall (n : nat) (x : R),
derive_pt (fun y : R => y ^ n) x
  (derivable_pt_pow n x) = INR n * x ^ Init.Nat.pred n
  intros; apply derive_pt_eq_0.n:nat
x:R
derivable_pt_lim (fun y : R => y ^ n) x
  (INR n * x ^ Init.Nat.pred n)
  apply derivable_pt_lim_pow.
Qed.

Lemma pr_nu :
  forall f (x:R) (pr1 pr2:derivable_pt f x),
    derive_pt f x pr1 = derive_pt f x pr2.forall (f : R -> R) (x : R)
  (pr1 pr2 : derivable_pt f x),
derive_pt f x pr1 = derive_pt f x pr2
Proof.forall (f : R -> R) (x : R)
  (pr1 pr2 : derivable_pt f x),
derive_pt f x pr1 = derive_pt f x pr2
  intros f x (x0,H0) (x1,H1).f:R -> R
x, x0:R
H0:derivable_pt_abs f x x0
x1:R
H1:derivable_pt_abs f x x1
derive_pt f x
  (exist (fun l : R => derivable_pt_abs f x l) x0 H0) =
derive_pt f x
  (exist (fun l : R => derivable_pt_abs f x l) x1 H1)
  apply (uniqueness_limite f x x0 x1 H0 H1).
Qed.


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

Local extremum's condition

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

Theorem deriv_maximum :
  forall f (a b c:R) (pr:derivable_pt f c),
    a < c ->
    c < b ->
    (forall x:R, a < x -> x < b -> f x <= f c) -> derive_pt f c pr = 0.forall (f : R -> R) (a b c : R)
  (pr : derivable_pt f c),
a < c ->
c < b ->
(forall x : R, a < x -> x < b -> f x <= f c) ->
derive_pt f c pr = 0
Proof.forall (f : R -> R) (a b c : R)
  (pr : derivable_pt f c),
a < c ->
c < b ->
(forall x : R, a < x -> x < b -> f x <= f c) ->
derive_pt f c pr = 0
  intros; case (Rtotal_order 0 (derive_pt f c pr)); intro.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 < derive_pt f c pr
derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  assert (H3 := derivable_derive f c pr).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 < derive_pt f c pr
H3:exists l : R, derive_pt f c pr = l
derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  elim H3; intros l H4; rewrite H4 in H2.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  assert (H5 := derive_pt_eq_1 f c l pr H4).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  cut (0 < l / 2);
    [ intro
      | unfold Rdiv; apply Rmult_lt_0_compat;
        [ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ].f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  elim (H5 (l / 2) H6); intros delta H7.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  cut (0 < (b - c) / 2).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2 -> derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  intro; cut (Rmin (delta / 2) ((b - c) / 2) <> 0).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0 ->
derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  intro; cut (Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta ->
derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  intro.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  assert (H11 := H7 (Rmin (delta / 2) ((b - c) / 2)) H9 H10).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  cut (0 < Rmin (delta / 2) ((b - c) / 2)).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2) ->
derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  intro; cut (a < c + Rmin (delta / 2) ((b - c) / 2)).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2) ->
derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  intro; cut (c + Rmin (delta / 2) ((b - c) / 2) < b).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b ->
derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  intro; assert (H15 := H1 (c + Rmin (delta / 2) ((b - c) / 2)) H13 H14).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  cut
    ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
      Rmin (delta / 2) ((b - c) / 2) <= 0).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0 ->
derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  intro; cut (- l < 0).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
- l < 0 -> derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
- l < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  intro; unfold Rminus in H11.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
- l < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  cut
    ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
      Rmin (delta / 2) ((b + - c) / 2) + - l < 0).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
(f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0 -> derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
(f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
- l < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  intro;
    cut
      (Rabs
        ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
          Rmin (delta / 2) ((b + - c) / 2) + - l) < l / 2).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
H18:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
   Rmin (delta / 2) ((b + - c) / 2) + 
   - l) < l / 2 -> derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
H18:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
   Rmin (delta / 2) ((b + - c) / 2) + 
   - l) < l / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
(f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
- l < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  unfold Rabs;
    case
    (Rcase_abs
      ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
        Rmin (delta / 2) ((b + - c) / 2) + - l)) as [Hlt|Hge].f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
H18, Hlt:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
-
((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
 Rmin (delta / 2) ((b + - c) / 2) + 
 - l) < l / 2 -> derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
H18:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
Hge:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l >= 0
(f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2) + 
- l < l / 2 -> derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
H18:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
   Rmin (delta / 2) ((b + - c) / 2) + 
   - l) < l / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
(f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
- l < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  replace
  (-
    ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
      Rmin (delta / 2) ((b + - c) / 2) + - l)) with
  (l +
    -
    ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
      Rmin (delta / 2) ((b + - c) / 2))).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
H18, Hlt:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
l +
-
((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
 Rmin (delta / 2) ((b + - c) / 2)) < 
l / 2 -> derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
H18, Hlt:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
l +
-
((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
 Rmin (delta / 2) ((b + - c) / 2)) =
-
((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
 Rmin (delta / 2) ((b + - c) / 2) + 
 - l)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
H18:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
Hge:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l >= 0
(f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2) + 
- l < l / 2 -> derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
H18:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
   Rmin (delta / 2) ((b + - c) / 2) + 
   - l) < l / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
(f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
- l < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  intro;
    generalize
      (Rplus_lt_compat_l (- l)
        (l +
          -
          ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
            Rmin (delta / 2) ((b + - c) / 2))) (l / 2) H19);
      repeat rewrite <- Rplus_assoc; rewrite Rplus_opp_l;
        rewrite Rplus_0_l; replace (- l + l / 2) with (- (l / 2)).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
H18, Hlt:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
H19:l +
-
((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
  - f c) / Rmin (delta / 2) ((b + - c) / 2)) <
l / 2
-
((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
 Rmin (delta / 2) ((b + - c) / 2)) < 
- (l / 2) -> derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
H18, Hlt:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
H19:l +
-
((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
  - f c) / Rmin (delta / 2) ((b + - c) / 2)) <
l / 2
- (l / 2) = - l + l / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
H18, Hlt:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
l +
-
((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
 Rmin (delta / 2) ((b + - c) / 2)) =
-
((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
 Rmin (delta / 2) ((b + - c) / 2) + 
 - l)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
H18:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
Hge:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l >= 0
(f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2) + 
- l < l / 2 -> derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
H18:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
   Rmin (delta / 2) ((b + - c) / 2) + 
   - l) < l / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
(f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
- l < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  intro;
    generalize
      (Ropp_lt_gt_contravar
        (-
          ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
            Rmin (delta / 2) ((b + - c) / 2))) (- (l / 2)) H20);
      repeat rewrite Ropp_involutive; intro;
        generalize
          (Rlt_trans 0 (l / 2)
            ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
              Rmin (delta / 2) ((b + - c) / 2)) H6 H21); intro;
          elim
            (Rlt_irrefl 0
              (Rlt_le_trans 0
                ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
                  Rmin (delta / 2) ((b + - c) / 2)) 0 H22 H16)).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
H18, Hlt:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
H19:l +
-
((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
  - f c) / Rmin (delta / 2) ((b + - c) / 2)) <
l / 2
- (l / 2) = - l + l / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
H18, Hlt:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
l +
-
((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
 Rmin (delta / 2) ((b + - c) / 2)) =
-
((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
 Rmin (delta / 2) ((b + - c) / 2) + 
 - l)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
H18:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
Hge:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l >= 0
(f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2) + 
- l < l / 2 -> derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
H18:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
   Rmin (delta / 2) ((b + - c) / 2) + 
   - l) < l / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
(f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
- l < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  pattern l at 2; rewrite double_var.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
H18, Hlt:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
H19:l +
-
((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
  - f c) / Rmin (delta / 2) ((b + - c) / 2)) <
l / 2
- (l / 2) = - (l / 2 + l / 2) + l / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
H18, Hlt:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
l +
-
((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
 Rmin (delta / 2) ((b + - c) / 2)) =
-
((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
 Rmin (delta / 2) ((b + - c) / 2) + 
 - l)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
H18:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
Hge:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l >= 0
(f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2) + 
- l < l / 2 -> derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
H18:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
   Rmin (delta / 2) ((b + - c) / 2) + 
   - l) < l / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
(f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
- l < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  ring.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
H18, Hlt:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
l +
-
((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
 Rmin (delta / 2) ((b + - c) / 2)) =
-
((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
 Rmin (delta / 2) ((b + - c) / 2) + 
 - l)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
H18:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
Hge:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l >= 0
(f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2) + 
- l < l / 2 -> derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
H18:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
   Rmin (delta / 2) ((b + - c) / 2) + 
   - l) < l / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
(f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
- l < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  ring.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
H18:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
Hge:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l >= 0
(f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2) + 
- l < l / 2 -> derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
H18:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
   Rmin (delta / 2) ((b + - c) / 2) + 
   - l) < l / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
(f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
- l < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  intro.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
H18:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
Hge:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l >= 0
H19:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < l / 2
derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
H18:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
   Rmin (delta / 2) ((b + - c) / 2) + 
   - l) < l / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
(f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
- l < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  assert
    (H20 :=
      Rge_le
      ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
        Rmin (delta / 2) ((b + - c) / 2) + - l) 0 Hge).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
H18:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
Hge:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l >= 0
H19:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < l / 2
H20:0 <=
(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l
derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
H18:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
   Rmin (delta / 2) ((b + - c) / 2) + 
   - l) < l / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
(f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
- l < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H20 H18)).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
H18:(f (c + Rmin (delta / 2) ((b + - c) / 2)) +
 - f c) / Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
   Rmin (delta / 2) ((b + - c) / 2) + 
   - l) < l / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
(f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
- l < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  assumption.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
(f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2) + 
- l < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
- l < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  rewrite <- Ropp_0;
    replace
    ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
      Rmin (delta / 2) ((b + - c) / 2) + - l) with
    (-
      (l +
        -
        ((f (c + Rmin (delta / 2) ((b + - c) / 2)) - f c) /
          Rmin (delta / 2) ((b + - c) / 2)))).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
-
(l +
 -
 ((f (c + Rmin (delta / 2) ((b + - c) / 2)) - f c) /
  Rmin (delta / 2) ((b + - c) / 2))) < 
- 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
-
(l +
 -
 ((f (c + Rmin (delta / 2) ((b + - c) / 2)) - f c) /
  Rmin (delta / 2) ((b + - c) / 2))) =
(f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2) + 
- l
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
- l < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  apply Ropp_gt_lt_contravar;
    change
      (0 <
        l +
        -
        ((f (c + Rmin (delta / 2) ((b + - c) / 2)) - f c) /
          Rmin (delta / 2) ((b + - c) / 2))); apply Rplus_lt_le_0_compat;
      [ assumption
        | rewrite <- Ropp_0; apply Ropp_ge_le_contravar; apply Rle_ge; assumption ].f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b + - c) / 2)) +
    - f c) / Rmin (delta / 2) ((b + - c) / 2) +
   - l) < l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
H17:- l < 0
-
(l +
 -
 ((f (c + Rmin (delta / 2) ((b + - c) / 2)) - f c) /
  Rmin (delta / 2) ((b + - c) / 2))) =
(f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
Rmin (delta / 2) ((b + - c) / 2) + 
- l
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
- l < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  unfold Rminus; ring.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
- l < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; assumption.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2) <= 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  replace
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
    Rmin (delta / 2) ((b - c) / 2)) with
  (-
    ((f c - f (c + Rmin (delta / 2) ((b - c) / 2))) /
      Rmin (delta / 2) ((b - c) / 2))).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
-
((f c - f (c + Rmin (delta / 2) ((b - c) / 2))) /
 Rmin (delta / 2) ((b - c) / 2)) <= 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
-
((f c - f (c + Rmin (delta / 2) ((b - c) / 2))) /
 Rmin (delta / 2) ((b - c) / 2)) =
(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  rewrite <- Ropp_0; apply Ropp_ge_le_contravar; apply Rle_ge;
    unfold Rdiv; apply Rmult_le_pos;
      [ generalize
        (Rplus_le_compat_r (- f (c + Rmin (delta * / 2) ((b - c) * / 2)))
          (f (c + Rmin (delta * / 2) ((b - c) * / 2))) (
            f c) H15); rewrite Rplus_opp_r; intro; assumption
        | left; apply Rinv_0_lt_compat; assumption ].f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
-
((f c - f (c + Rmin (delta / 2) ((b - c) / 2))) /
 Rmin (delta / 2) ((b - c) / 2)) =
(f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  unfold Rdiv.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
-
((f c - f (c + Rmin (delta * / 2) ((b - c) * / 2))) *
 / Rmin (delta * / 2) ((b - c) * / 2)) =
(f (c + Rmin (delta * / 2) ((b - c) * / 2)) - f c) *
/ Rmin (delta * / 2) ((b - c) * / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  rewrite <- Ropp_mult_distr_l_reverse.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
- (f c - f (c + Rmin (delta * / 2) ((b - c) * / 2))) *
/ Rmin (delta * / 2) ((b - c) * / 2) =
(f (c + Rmin (delta * / 2) ((b - c) * / 2)) - f c) *
/ Rmin (delta * / 2) ((b - c) * / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  repeat rewrite <- (Rmult_comm (/ Rmin (delta * / 2) ((b - c) * / 2))).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
/ Rmin (delta * / 2) ((b - c) * / 2) *
- (f c - f (c + Rmin (delta * / 2) ((b - c) * / 2))) =
/ Rmin (delta * / 2) ((b - c) * / 2) *
(f (c + Rmin (delta * / 2) ((b - c) * / 2)) - f c)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  apply Rmult_eq_reg_l with (Rmin (delta * / 2) ((b - c) * / 2)).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
Rmin (delta * / 2) ((b - c) * / 2) *
(/ Rmin (delta * / 2) ((b - c) * / 2) *
 - (f c - f (c + Rmin (delta * / 2) ((b - c) * / 2)))) =
Rmin (delta * / 2) ((b - c) * / 2) *
(/ Rmin (delta * / 2) ((b - c) * / 2) *
 (f (c + Rmin (delta * / 2) ((b - c) * / 2)) - f c))
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
Rmin (delta * / 2) ((b - c) * / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  repeat rewrite <- Rmult_assoc.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
Rmin (delta * / 2) ((b - c) * / 2) *
/ Rmin (delta * / 2) ((b - c) * / 2) *
- (f c - f (c + Rmin (delta * / 2) ((b - c) * / 2))) =
Rmin (delta * / 2) ((b - c) * / 2) *
/ Rmin (delta * / 2) ((b - c) * / 2) *
(f (c + Rmin (delta * / 2) ((b - c) * / 2)) - f c)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
Rmin (delta * / 2) ((b - c) * / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  rewrite <- Rinv_r_sym.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
1 *
- (f c - f (c + Rmin (delta * / 2) ((b - c) * / 2))) =
1 * (f (c + Rmin (delta * / 2) ((b - c) * / 2)) - f c)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
Rmin (delta * / 2) ((b - c) * / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
Rmin (delta * / 2) ((b - c) * / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  repeat rewrite Rmult_1_l.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
- (f c - f (c + Rmin (delta * / 2) ((b - c) * / 2))) =
f (c + Rmin (delta * / 2) ((b - c) * / 2)) - f c
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
Rmin (delta * / 2) ((b - c) * / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
Rmin (delta * / 2) ((b - c) * / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  ring.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
Rmin (delta * / 2) ((b - c) * / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
Rmin (delta * / 2) ((b - c) * / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  red; intro.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:Rmin (delta * / 2) ((b - c) * / 2) = 0
False
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
Rmin (delta * / 2) ((b - c) * / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  unfold Rdiv in H12; rewrite H16 in H12; elim (Rlt_irrefl 0 H12).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
Rmin (delta * / 2) ((b - c) * / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  red; intro.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:c + Rmin (delta / 2) ((b - c) / 2) < b
H15:f (c + Rmin (delta / 2) ((b - c) / 2)) <= f c
H16:Rmin (delta * / 2) ((b - c) * / 2) = 0
False
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  unfold Rdiv in H12; rewrite H16 in H12; elim (Rlt_irrefl 0 H12).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  assert (H14 := Rmin_r (delta / 2) ((b - c) / 2)).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:Rmin (delta / 2) ((b - c) / 2) <= (b - c) / 2
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  assert
    (H15 :=
      Rplus_le_compat_l c (Rmin (delta / 2) ((b - c) / 2)) ((b - c) / 2) H14).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:Rmin (delta / 2) ((b - c) / 2) <= (b - c) / 2
H15:c + Rmin (delta / 2) ((b - c) / 2) <=
c + (b - c) / 2
c + Rmin (delta / 2) ((b - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  apply Rle_lt_trans with (c + (b - c) / 2).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:Rmin (delta / 2) ((b - c) / 2) <= (b - c) / 2
H15:c + Rmin (delta / 2) ((b - c) / 2) <=
c + (b - c) / 2
c + Rmin (delta / 2) ((b - c) / 2) <= c + (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:Rmin (delta / 2) ((b - c) / 2) <= (b - c) / 2
H15:c + Rmin (delta / 2) ((b - c) / 2) <=
c + (b - c) / 2
c + (b - c) / 2 < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  assumption.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:Rmin (delta / 2) ((b - c) / 2) <= (b - c) / 2
H15:c + Rmin (delta / 2) ((b - c) / 2) <=
c + (b - c) / 2
c + (b - c) / 2 < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  apply Rmult_lt_reg_l with 2.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:Rmin (delta / 2) ((b - c) / 2) <= (b - c) / 2
H15:c + Rmin (delta / 2) ((b - c) / 2) <=
c + (b - c) / 2
0 < 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:Rmin (delta / 2) ((b - c) / 2) <= (b - c) / 2
H15:c + Rmin (delta / 2) ((b - c) / 2) <=
c + (b - c) / 2
2 * (c + (b - c) / 2) < 2 * b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  prove_sup0.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:Rmin (delta / 2) ((b - c) / 2) <= (b - c) / 2
H15:c + Rmin (delta / 2) ((b - c) / 2) <=
c + (b - c) / 2
2 * (c + (b - c) / 2) < 2 * b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  replace (2 * (c + (b - c) / 2)) with (c + b).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:Rmin (delta / 2) ((b - c) / 2) <= (b - c) / 2
H15:c + Rmin (delta / 2) ((b - c) / 2) <=
c + (b - c) / 2
c + b < 2 * b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:Rmin (delta / 2) ((b - c) / 2) <= (b - c) / 2
H15:c + Rmin (delta / 2) ((b - c) / 2) <=
c + (b - c) / 2
c + b = 2 * (c + (b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  replace (2 * b) with (b + b).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:Rmin (delta / 2) ((b - c) / 2) <= (b - c) / 2
H15:c + Rmin (delta / 2) ((b - c) / 2) <=
c + (b - c) / 2
c + b < b + b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:Rmin (delta / 2) ((b - c) / 2) <= (b - c) / 2
H15:c + Rmin (delta / 2) ((b - c) / 2) <=
c + (b - c) / 2
b + b = 2 * b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:Rmin (delta / 2) ((b - c) / 2) <= (b - c) / 2
H15:c + Rmin (delta / 2) ((b - c) / 2) <=
c + (b - c) / 2
c + b = 2 * (c + (b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  apply Rplus_lt_compat_r; assumption.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:Rmin (delta / 2) ((b - c) / 2) <= (b - c) / 2
H15:c + Rmin (delta / 2) ((b - c) / 2) <=
c + (b - c) / 2
b + b = 2 * b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:Rmin (delta / 2) ((b - c) / 2) <= (b - c) / 2
H15:c + Rmin (delta / 2) ((b - c) / 2) <=
c + (b - c) / 2
c + b = 2 * (c + (b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  ring.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:Rmin (delta / 2) ((b - c) / 2) <= (b - c) / 2
H15:c + Rmin (delta / 2) ((b - c) / 2) <=
c + (b - c) / 2
c + b = 2 * (c + (b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  unfold Rdiv; rewrite Rmult_plus_distr_l.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:Rmin (delta / 2) ((b - c) / 2) <= (b - c) / 2
H15:c + Rmin (delta / 2) ((b - c) / 2) <=
c + (b - c) / 2
c + b = 2 * c + 2 * ((b - c) * / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  repeat rewrite (Rmult_comm 2).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:Rmin (delta / 2) ((b - c) / 2) <= (b - c) / 2
H15:c + Rmin (delta / 2) ((b - c) / 2) <=
c + (b - c) / 2
c + b = c * 2 + (b - c) * / 2 * 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  rewrite Rmult_assoc; rewrite <- Rinv_l_sym.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:Rmin (delta / 2) ((b - c) / 2) <= (b - c) / 2
H15:c + Rmin (delta / 2) ((b - c) / 2) <=
c + (b - c) / 2
c + b = c * 2 + (b - c) * 1
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:Rmin (delta / 2) ((b - c) / 2) <= (b - c) / 2
H15:c + Rmin (delta / 2) ((b - c) / 2) <=
c + (b - c) / 2
2 <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  rewrite Rmult_1_r.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:Rmin (delta / 2) ((b - c) / 2) <= (b - c) / 2
H15:c + Rmin (delta / 2) ((b - c) / 2) <=
c + (b - c) / 2
c + b = c * 2 + (b - c)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:Rmin (delta / 2) ((b - c) / 2) <= (b - c) / 2
H15:c + Rmin (delta / 2) ((b - c) / 2) <=
c + (b - c) / 2
2 <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  ring.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
H13:a < c + Rmin (delta / 2) ((b - c) / 2)
H14:Rmin (delta / 2) ((b - c) / 2) <= (b - c) / 2
H15:c + Rmin (delta / 2) ((b - c) / 2) <=
c + (b - c) / 2
2 <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  discrR.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  apply Rlt_trans with c.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
a < c
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
c < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  assumption.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
H12:0 < Rmin (delta / 2) ((b - c) / 2)
c < c + Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  pattern c at 1; rewrite <- (Rplus_0_r c); apply Rplus_lt_compat_l;
    assumption.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  cut (0 < delta / 2).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < delta / 2 -> 0 < Rmin (delta / 2) ((b - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < delta / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  intro;
    apply
      (Rmin_stable_in_posreal (mkposreal (delta / 2) H12)
        (mkposreal ((b - c) / 2) H8)).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
H10:Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
H11:Rabs
  ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) /
   Rmin (delta / 2) ((b - c) / 2) - l) < 
l / 2
0 < delta / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  unfold Rdiv; apply Rmult_lt_0_compat;
    [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ].f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  unfold Rabs; case (Rcase_abs (Rmin (delta / 2) ((b - c) / 2))) as [Hlt|Hge].f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Hlt:Rmin (delta / 2) ((b - c) / 2) < 0
- Rmin (delta / 2) ((b - c) / 2) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Hge:Rmin (delta / 2) ((b - c) / 2) >= 0
Rmin (delta / 2) ((b - c) / 2) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  cut (0 < delta / 2).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Hlt:Rmin (delta / 2) ((b - c) / 2) < 0
0 < delta / 2 ->
- Rmin (delta / 2) ((b - c) / 2) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Hlt:Rmin (delta / 2) ((b - c) / 2) < 0
0 < delta / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Hge:Rmin (delta / 2) ((b - c) / 2) >= 0
Rmin (delta / 2) ((b - c) / 2) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  intro.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Hlt:Rmin (delta / 2) ((b - c) / 2) < 0
H10:0 < delta / 2
- Rmin (delta / 2) ((b - c) / 2) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Hlt:Rmin (delta / 2) ((b - c) / 2) < 0
0 < delta / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Hge:Rmin (delta / 2) ((b - c) / 2) >= 0
Rmin (delta / 2) ((b - c) / 2) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  generalize
    (Rmin_stable_in_posreal (mkposreal (delta / 2) H10)
      (mkposreal ((b - c) / 2) H8)); simpl; intro;
    elim (Rlt_irrefl 0 (Rlt_trans 0 (Rmin (delta / 2) ((b - c) / 2)) 0 H11 Hlt)).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Hlt:Rmin (delta / 2) ((b - c) / 2) < 0
0 < delta / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Hge:Rmin (delta / 2) ((b - c) / 2) >= 0
Rmin (delta / 2) ((b - c) / 2) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  unfold Rdiv; apply Rmult_lt_0_compat;
    [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ].f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Hge:Rmin (delta / 2) ((b - c) / 2) >= 0
Rmin (delta / 2) ((b - c) / 2) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  apply Rle_lt_trans with (delta / 2).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Hge:Rmin (delta / 2) ((b - c) / 2) >= 0
Rmin (delta / 2) ((b - c) / 2) <= delta / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Hge:Rmin (delta / 2) ((b - c) / 2) >= 0
delta / 2 < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  apply Rmin_l.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Hge:Rmin (delta / 2) ((b - c) / 2) >= 0
delta / 2 < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  unfold Rdiv; apply Rmult_lt_reg_l with 2.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Hge:Rmin (delta / 2) ((b - c) / 2) >= 0
0 < 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Hge:Rmin (delta / 2) ((b - c) / 2) >= 0
2 * (delta * / 2) < 2 * delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  prove_sup0.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Hge:Rmin (delta / 2) ((b - c) / 2) >= 0
2 * (delta * / 2) < 2 * delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Hge:Rmin (delta / 2) ((b - c) / 2) >= 0
1 * delta < 2 * delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Hge:Rmin (delta / 2) ((b - c) / 2) >= 0
2 <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  rewrite Rmult_1_l.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Hge:Rmin (delta / 2) ((b - c) / 2) >= 0
delta < 2 * delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Hge:Rmin (delta / 2) ((b - c) / 2) >= 0
2 <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  replace (2 * delta) with (delta + delta).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Hge:Rmin (delta / 2) ((b - c) / 2) >= 0
delta < delta + delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Hge:Rmin (delta / 2) ((b - c) / 2) >= 0
delta + delta = 2 * delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Hge:Rmin (delta / 2) ((b - c) / 2) >= 0
2 <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  pattern delta at 2; rewrite <- (Rplus_0_r delta);
    apply Rplus_lt_compat_l.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Hge:Rmin (delta / 2) ((b - c) / 2) >= 0
0 < delta + 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Hge:Rmin (delta / 2) ((b - c) / 2) >= 0
delta + delta = 2 * delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Hge:Rmin (delta / 2) ((b - c) / 2) >= 0
2 <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  rewrite Rplus_0_r; apply (cond_pos delta).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Hge:Rmin (delta / 2) ((b - c) / 2) >= 0
delta + delta = 2 * delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Hge:Rmin (delta / 2) ((b - c) / 2) >= 0
2 <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  symmetry ; apply double.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
H9:Rmin (delta / 2) ((b - c) / 2) <> 0
Hge:Rmin (delta / 2) ((b - c) / 2) >= 0
2 <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  discrR.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  cut (0 < delta / 2).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
0 < delta / 2 -> Rmin (delta / 2) ((b - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
0 < delta / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  intro;
    generalize
      (Rmin_stable_in_posreal (mkposreal (delta / 2) H9)
        (mkposreal ((b - c) / 2) H8)); simpl;
      intro; red; intro; rewrite H11 in H10; elim (Rlt_irrefl 0 H10).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
H8:0 < (b - c) / 2
0 < delta / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  unfold Rdiv; apply Rmult_lt_0_compat;
    [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ].f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < (b - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  unfold Rdiv; apply Rmult_lt_0_compat.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < b - c
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  generalize (Rplus_lt_compat_r (- c) c b H0); rewrite Rplus_opp_r; intro;
    assumption.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
l:R
H2:0 < l
H3:exists l0 : R, derive_pt f c pr = l0
H4:derive_pt f c pr = l
H5:derivable_pt_lim f c l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < l / 2
0 < / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  apply Rinv_0_lt_compat; prove_sup0.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
derive_pt f c pr = 0
  elim H2; intro.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
H3:0 = derive_pt f c pr
derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
H3:0 > derive_pt f c pr
derive_pt f c pr = 0
  symmetry ; assumption.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
H3:0 > derive_pt f c pr
derive_pt f c pr = 0
  generalize (derivable_derive f c pr); intro; elim H4; intros l H5.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
H3:0 > derive_pt f c pr
H4:exists l0 : R, derive_pt f c pr = l0
l:R
H5:derive_pt f c pr = l
derive_pt f c pr = 0
  rewrite H5 in H3; generalize (derive_pt_eq_1 f c l pr H5); intro;
    cut (0 < - (l / 2)).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2) -> derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  intro; elim (H6 (- (l / 2)) H7); intros delta H9.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  cut (0 < (c - a) / 2).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2 -> derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  intro; cut (Rmax (- (delta / 2)) ((a - c) / 2) < 0).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0 ->
derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  intro; cut (Rmax (- (delta / 2)) ((a - c) / 2) <> 0).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0 ->
derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  intro; cut (Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta ->
derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  intro; generalize (H9 (Rmax (- (delta / 2)) ((a - c) / 2)) H11 H12); intro;
    cut (a < c + Rmax (- (delta / 2)) ((a - c) / 2)).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
a < c + Rmax (- (delta / 2)) ((a - c) / 2) ->
derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
a < c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  cut (c + Rmax (- (delta / 2)) ((a - c) / 2) < b).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
c + Rmax (- (delta / 2)) ((a - c) / 2) < b ->
a < c + Rmax (- (delta / 2)) ((a - c) / 2) ->
derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
c + Rmax (- (delta / 2)) ((a - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
a < c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  intros; generalize (H1 (c + Rmax (- (delta / 2)) ((a - c) / 2)) H15 H14);
    intro;
      cut
        (0 <=
          (f (c + Rmax (- (delta / 2)) ((a - c) / 2)) - f c) /
          Rmax (- (delta / 2)) ((a - c) / 2)).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) - f c) /
Rmax (- (delta / 2)) ((a - c) / 2) ->
derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) - f c) /
Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
c + Rmax (- (delta / 2)) ((a - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
a < c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  intro; cut (0 < - l).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
0 < - l -> derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
0 < - l
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) - f c) /
Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
c + Rmax (- (delta / 2)) ((a - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
a < c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  intro; unfold Rminus in H13;
    cut
      (0 <
        (f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
        Rmax (- (delta / 2)) ((a + - c) / 2) + - l).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
Rmax (- (delta / 2)) ((a + - c) / 2) + 
- l -> derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
Rmax (- (delta / 2)) ((a + - c) / 2) + 
- l
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
0 < - l
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) - f c) /
Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
c + Rmax (- (delta / 2)) ((a - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
a < c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  intro;
    cut
      (Rabs
        ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
          Rmax (- (delta / 2)) ((a + - c) / 2) + - l) <
        - (l / 2)).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
H19:0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l
Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
   - l) < - (l / 2) -> derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
H19:0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l
Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
   - l) < - (l / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
Rmax (- (delta / 2)) ((a + - c) / 2) + 
- l
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
0 < - l
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) - f c) /
Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
c + Rmax (- (delta / 2)) ((a - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
a < c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  unfold Rabs;
    case
    (Rcase_abs
      ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
        Rmax (- (delta / 2)) ((a + - c) / 2) + - l)) as [Hlt|Hge].f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
H19:0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l
Hlt:(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l < 0
-
((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
 Rmax (- (delta / 2)) ((a + - c) / 2) + 
 - l) < - (l / 2) -> derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
H19:0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l
Hge:(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l >= 0
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
Rmax (- (delta / 2)) ((a + - c) / 2) + 
- l < - (l / 2) -> derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
H19:0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l
Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
   - l) < - (l / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
Rmax (- (delta / 2)) ((a + - c) / 2) + 
- l
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
0 < - l
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) - f c) /
Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
c + Rmax (- (delta / 2)) ((a - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
a < c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  elim
      (Rlt_irrefl 0
        (Rlt_trans 0
          ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
            Rmax (- (delta / 2)) ((a + - c) / 2) + - l) 0 H19 Hlt)).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
H19:0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l
Hge:(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l >= 0
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
Rmax (- (delta / 2)) ((a + - c) / 2) + 
- l < - (l / 2) -> derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
H19:0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l
Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
   - l) < - (l / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
Rmax (- (delta / 2)) ((a + - c) / 2) + 
- l
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
0 < - l
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) - f c) /
Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
c + Rmax (- (delta / 2)) ((a - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
a < c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  intros;
    generalize
      (Rplus_lt_compat_r l
        ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
          Rmax (- (delta / 2)) ((a + - c) / 2) + - l) (
            - (l / 2)) H20); repeat rewrite Rplus_assoc; rewrite Rplus_opp_l;
      rewrite Rplus_0_r; replace (- (l / 2) + l) with (l / 2).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
H19:0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l
Hge:(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l >= 0
H20:(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l < - (l / 2)
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
Rmax (- (delta / 2)) ((a + - c) / 2) < 
l / 2 -> derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
H19:0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l
Hge:(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l >= 0
H20:(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l < - (l / 2)
l / 2 = - (l / 2) + l
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
H19:0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l
Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
   - l) < - (l / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
Rmax (- (delta / 2)) ((a + - c) / 2) + 
- l
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
0 < - l
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) - f c) /
Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
c + Rmax (- (delta / 2)) ((a - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
a < c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  cut (l / 2 < 0).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
H19:0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l
Hge:(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l >= 0
H20:(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l < - (l / 2)
l / 2 < 0 ->
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
Rmax (- (delta / 2)) ((a + - c) / 2) < 
l / 2 -> derive_pt f c pr = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
H19:0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l
Hge:(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l >= 0
H20:(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l < - (l / 2)
l / 2 < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
H19:0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l
Hge:(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l >= 0
H20:(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l < - (l / 2)
l / 2 = - (l / 2) + l
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
H19:0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l
Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
   - l) < - (l / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
Rmax (- (delta / 2)) ((a + - c) / 2) + 
- l
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
0 < - l
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) - f c) /
Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
c + Rmax (- (delta / 2)) ((a - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
a < c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  intros;
    generalize
      (Rlt_trans
        ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
          Rmax (- (delta / 2)) ((a + - c) / 2)) (l / 2) 0 H22 H21);
      intro;
        elim
          (Rlt_irrefl 0
            (Rle_lt_trans 0
              ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) - f c) /
                Rmax (- (delta / 2)) ((a - c) / 2)) 0 H17 H23)).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
H19:0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l
Hge:(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l >= 0
H20:(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l < - (l / 2)
l / 2 < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
H19:0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l
Hge:(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l >= 0
H20:(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l < - (l / 2)
l / 2 = - (l / 2) + l
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
H19:0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l
Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
   - l) < - (l / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
Rmax (- (delta / 2)) ((a + - c) / 2) + 
- l
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
0 < - l
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) - f c) /
Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
c + Rmax (- (delta / 2)) ((a - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
a < c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  rewrite <- (Ropp_involutive (l / 2)); rewrite <- Ropp_0;
    apply Ropp_lt_gt_contravar; assumption.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
H19:0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l
Hge:(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l >= 0
H20:(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l < - (l / 2)
l / 2 = - (l / 2) + l
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
H19:0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l
Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
   - l) < - (l / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
Rmax (- (delta / 2)) ((a + - c) / 2) + 
- l
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
0 < - l
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) - f c) /
Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
c + Rmax (- (delta / 2)) ((a - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
a < c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  pattern l at 3; rewrite double_var.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
H19:0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l
Hge:(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l >= 0
H20:(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l < - (l / 2)
l / 2 = - (l / 2) + (l / 2 + l / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
H19:0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l
Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
   - l) < - (l / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
Rmax (- (delta / 2)) ((a + - c) / 2) + 
- l
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
0 < - l
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) - f c) /
Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
c + Rmax (- (delta / 2)) ((a - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
a < c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  ring.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
H19:0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
 - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
- l
Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) +
   - l) < - (l / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
Rmax (- (delta / 2)) ((a + - c) / 2) + 
- l
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
0 < - l
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) - f c) /
Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
c + Rmax (- (delta / 2)) ((a - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
a < c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  assumption.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) +
    - f c) /
   Rmax (- (delta / 2)) ((a + - c) / 2) + 
   - l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
H18:0 < - l
0 <
(f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
Rmax (- (delta / 2)) ((a + - c) / 2) + 
- l
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
0 < - l
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) - f c) /
Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
c + Rmax (- (delta / 2)) ((a - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
a < c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  apply Rplus_le_lt_0_compat; assumption.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
H17:0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
 f c) / Rmax (- (delta / 2)) ((a - c) / 2)
0 < - l
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) - f c) /
Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
c + Rmax (- (delta / 2)) ((a - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
a < c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; assumption.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
0 <=
(f (c + Rmax (- (delta / 2)) ((a - c) / 2)) - f c) /
Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
c + Rmax (- (delta / 2)) ((a - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
a < c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  unfold Rdiv;
    replace
    ((f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c) *
      / Rmax (- (delta * / 2)) ((a - c) * / 2)) with
    (- (f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c) *
      / - Rmax (- (delta * / 2)) ((a - c) * / 2)).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
0 <=
-
(f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c) *
/ - Rmax (- (delta * / 2)) ((a - c) * / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
-
(f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c) *
/ - Rmax (- (delta * / 2)) ((a - c) * / 2) =
(f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c) *
/ Rmax (- (delta * / 2)) ((a - c) * / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
c + Rmax (- (delta / 2)) ((a - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
a < c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  apply Rmult_le_pos.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
0 <=
-
(f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
0 <= / - Rmax (- (delta * / 2)) ((a - c) * / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
-
(f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c) *
/ - Rmax (- (delta * / 2)) ((a - c) * / 2) =
(f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c) *
/ Rmax (- (delta * / 2)) ((a - c) * / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
c + Rmax (- (delta / 2)) ((a - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
a < c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  generalize
    (Rplus_le_compat_l (- f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)))
      (f (c + Rmax (- (delta * / 2)) ((a - c) * / 2))) (
        f c) H16); rewrite Rplus_opp_l;
    replace (- (f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c)) with
    (- f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) + f c).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
0 <=
- f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) + f c ->
0 <=
- f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) + f c
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
- f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) + f c =
-
(f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
0 <= / - Rmax (- (delta * / 2)) ((a - c) * / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
-
(f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c) *
/ - Rmax (- (delta * / 2)) ((a - c) * / 2) =
(f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c) *
/ Rmax (- (delta * / 2)) ((a - c) * / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
c + Rmax (- (delta / 2)) ((a - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
a < c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  intro; assumption.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
- f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) + f c =
-
(f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
0 <= / - Rmax (- (delta * / 2)) ((a - c) * / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
-
(f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c) *
/ - Rmax (- (delta * / 2)) ((a - c) * / 2) =
(f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c) *
/ Rmax (- (delta * / 2)) ((a - c) * / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
c + Rmax (- (delta / 2)) ((a - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
a < c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  ring.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
0 <= / - Rmax (- (delta * / 2)) ((a - c) * / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
-
(f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c) *
/ - Rmax (- (delta * / 2)) ((a - c) * / 2) =
(f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c) *
/ Rmax (- (delta * / 2)) ((a - c) * / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
c + Rmax (- (delta / 2)) ((a - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
a < c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  left; apply Rinv_0_lt_compat; rewrite <- Ropp_0; apply Ropp_lt_gt_contravar;
    assumption.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
-
(f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c) *
/ - Rmax (- (delta * / 2)) ((a - c) * / 2) =
(f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c) *
/ Rmax (- (delta * / 2)) ((a - c) * / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
c + Rmax (- (delta / 2)) ((a - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
a < c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  unfold Rdiv.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
-
(f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c) *
/ - Rmax (- (delta * / 2)) ((a - c) * / 2) =
(f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c) *
/ Rmax (- (delta * / 2)) ((a - c) * / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
c + Rmax (- (delta / 2)) ((a - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
a < c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  rewrite <- Ropp_inv_permute.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
-
(f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c) *
- / Rmax (- (delta * / 2)) ((a - c) * / 2) =
(f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c) *
/ Rmax (- (delta * / 2)) ((a - c) * / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
Rmax (- (delta * / 2)) ((a - c) * / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
c + Rmax (- (delta / 2)) ((a - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
a < c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  rewrite Rmult_opp_opp.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
(f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c) *
/ Rmax (- (delta * / 2)) ((a - c) * / 2) =
(f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c) *
/ Rmax (- (delta * / 2)) ((a - c) * / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
Rmax (- (delta * / 2)) ((a - c) * / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
c + Rmax (- (delta / 2)) ((a - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
a < c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  reflexivity.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:c + Rmax (- (delta / 2)) ((a - c) / 2) < b
H15:a < c + Rmax (- (delta / 2)) ((a - c) / 2)
H16:f (c + Rmax (- (delta / 2)) ((a - c) / 2)) <=
f c
Rmax (- (delta * / 2)) ((a - c) * / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
c + Rmax (- (delta / 2)) ((a - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
a < c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  unfold Rdiv in H11; assumption.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
c + Rmax (- (delta / 2)) ((a - c) / 2) < b
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
a < c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  generalize (Rplus_lt_compat_l c (Rmax (- (delta / 2)) ((a - c) / 2)) 0 H10);
    rewrite Rplus_0_r; intro; apply Rlt_trans with c;
      assumption.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
a < c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  generalize (RmaxLess2 (- (delta / 2)) ((a - c) / 2)); intro;
    generalize
      (Rplus_le_compat_l c ((a - c) / 2) (Rmax (- (delta / 2)) ((a - c) / 2)) H14);
      intro; apply Rlt_le_trans with (c + (a - c) / 2).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:(a - c) / 2 <=
Rmax (- (delta / 2)) ((a - c) / 2)
H15:c + (a - c) / 2 <=
c + Rmax (- (delta / 2)) ((a - c) / 2)
a < c + (a - c) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:(a - c) / 2 <=
Rmax (- (delta / 2)) ((a - c) / 2)
H15:c + (a - c) / 2 <=
c + Rmax (- (delta / 2)) ((a - c) / 2)
c + (a - c) / 2 <=
c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  apply Rmult_lt_reg_l with 2.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:(a - c) / 2 <=
Rmax (- (delta / 2)) ((a - c) / 2)
H15:c + (a - c) / 2 <=
c + Rmax (- (delta / 2)) ((a - c) / 2)
0 < 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:(a - c) / 2 <=
Rmax (- (delta / 2)) ((a - c) / 2)
H15:c + (a - c) / 2 <=
c + Rmax (- (delta / 2)) ((a - c) / 2)
2 * a < 2 * (c + (a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:(a - c) / 2 <=
Rmax (- (delta / 2)) ((a - c) / 2)
H15:c + (a - c) / 2 <=
c + Rmax (- (delta / 2)) ((a - c) / 2)
c + (a - c) / 2 <=
c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  prove_sup0.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:(a - c) / 2 <=
Rmax (- (delta / 2)) ((a - c) / 2)
H15:c + (a - c) / 2 <=
c + Rmax (- (delta / 2)) ((a - c) / 2)
2 * a < 2 * (c + (a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:(a - c) / 2 <=
Rmax (- (delta / 2)) ((a - c) / 2)
H15:c + (a - c) / 2 <=
c + Rmax (- (delta / 2)) ((a - c) / 2)
c + (a - c) / 2 <=
c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  replace (2 * (c + (a - c) / 2)) with (a + c).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:(a - c) / 2 <=
Rmax (- (delta / 2)) ((a - c) / 2)
H15:c + (a - c) / 2 <=
c + Rmax (- (delta / 2)) ((a - c) / 2)
2 * a < a + c
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:(a - c) / 2 <=
Rmax (- (delta / 2)) ((a - c) / 2)
H15:c + (a - c) / 2 <=
c + Rmax (- (delta / 2)) ((a - c) / 2)
a + c = 2 * (c + (a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:(a - c) / 2 <=
Rmax (- (delta / 2)) ((a - c) / 2)
H15:c + (a - c) / 2 <=
c + Rmax (- (delta / 2)) ((a - c) / 2)
c + (a - c) / 2 <=
c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  rewrite double.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:(a - c) / 2 <=
Rmax (- (delta / 2)) ((a - c) / 2)
H15:c + (a - c) / 2 <=
c + Rmax (- (delta / 2)) ((a - c) / 2)
a + a < a + c
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:(a - c) / 2 <=
Rmax (- (delta / 2)) ((a - c) / 2)
H15:c + (a - c) / 2 <=
c + Rmax (- (delta / 2)) ((a - c) / 2)
a + c = 2 * (c + (a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:(a - c) / 2 <=
Rmax (- (delta / 2)) ((a - c) / 2)
H15:c + (a - c) / 2 <=
c + Rmax (- (delta / 2)) ((a - c) / 2)
c + (a - c) / 2 <=
c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  apply Rplus_lt_compat_l; assumption.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:(a - c) / 2 <=
Rmax (- (delta / 2)) ((a - c) / 2)
H15:c + (a - c) / 2 <=
c + Rmax (- (delta / 2)) ((a - c) / 2)
a + c = 2 * (c + (a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:(a - c) / 2 <=
Rmax (- (delta / 2)) ((a - c) / 2)
H15:c + (a - c) / 2 <=
c + Rmax (- (delta / 2)) ((a - c) / 2)
c + (a - c) / 2 <=
c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  field; discrR.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
H12:Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) <
delta
H13:Rabs
  ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) -
    f c) / Rmax (- (delta / 2)) ((a - c) / 2) -
   l) < - (l / 2)
H14:(a - c) / 2 <=
Rmax (- (delta / 2)) ((a - c) / 2)
H15:c + (a - c) / 2 <=
c + Rmax (- (delta / 2)) ((a - c) / 2)
c + (a - c) / 2 <=
c + Rmax (- (delta / 2)) ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  assumption.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  unfold Rabs; case (Rcase_abs (Rmax (- (delta / 2)) ((a - c) / 2))) as [Hlt|Hge].f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hlt:Rmax (- (delta / 2)) ((a - c) / 2) < 0
- Rmax (- (delta / 2)) ((a - c) / 2) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hge:Rmax (- (delta / 2)) ((a - c) / 2) >= 0
Rmax (- (delta / 2)) ((a - c) / 2) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  generalize (RmaxLess1 (- (delta / 2)) ((a - c) / 2)); intro;
    generalize
      (Ropp_le_ge_contravar (- (delta / 2)) (Rmax (- (delta / 2)) ((a - c) / 2))
        H12); rewrite Ropp_involutive; intro;
      generalize (Rge_le (delta / 2) (- Rmax (- (delta / 2)) ((a - c) / 2)) H13);
        intro; apply Rle_lt_trans with (delta / 2).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hlt:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H12:- (delta / 2) <=
Rmax (- (delta / 2)) ((a - c) / 2)
H13:delta / 2 >=
- Rmax (- (delta / 2)) ((a - c) / 2)
H14:- Rmax (- (delta / 2)) ((a - c) / 2) <=
delta / 2
- Rmax (- (delta / 2)) ((a - c) / 2) <= delta / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hlt:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H12:- (delta / 2) <=
Rmax (- (delta / 2)) ((a - c) / 2)
H13:delta / 2 >=
- Rmax (- (delta / 2)) ((a - c) / 2)
H14:- Rmax (- (delta / 2)) ((a - c) / 2) <=
delta / 2
delta / 2 < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hge:Rmax (- (delta / 2)) ((a - c) / 2) >= 0
Rmax (- (delta / 2)) ((a - c) / 2) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  assumption.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hlt:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H12:- (delta / 2) <=
Rmax (- (delta / 2)) ((a - c) / 2)
H13:delta / 2 >=
- Rmax (- (delta / 2)) ((a - c) / 2)
H14:- Rmax (- (delta / 2)) ((a - c) / 2) <=
delta / 2
delta / 2 < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hge:Rmax (- (delta / 2)) ((a - c) / 2) >= 0
Rmax (- (delta / 2)) ((a - c) / 2) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  apply Rmult_lt_reg_l with 2.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hlt:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H12:- (delta / 2) <=
Rmax (- (delta / 2)) ((a - c) / 2)
H13:delta / 2 >=
- Rmax (- (delta / 2)) ((a - c) / 2)
H14:- Rmax (- (delta / 2)) ((a - c) / 2) <=
delta / 2
0 < 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hlt:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H12:- (delta / 2) <=
Rmax (- (delta / 2)) ((a - c) / 2)
H13:delta / 2 >=
- Rmax (- (delta / 2)) ((a - c) / 2)
H14:- Rmax (- (delta / 2)) ((a - c) / 2) <=
delta / 2
2 * (delta / 2) < 2 * delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hge:Rmax (- (delta / 2)) ((a - c) / 2) >= 0
Rmax (- (delta / 2)) ((a - c) / 2) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  prove_sup0.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hlt:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H12:- (delta / 2) <=
Rmax (- (delta / 2)) ((a - c) / 2)
H13:delta / 2 >=
- Rmax (- (delta / 2)) ((a - c) / 2)
H14:- Rmax (- (delta / 2)) ((a - c) / 2) <=
delta / 2
2 * (delta / 2) < 2 * delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hge:Rmax (- (delta / 2)) ((a - c) / 2) >= 0
Rmax (- (delta / 2)) ((a - c) / 2) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
    rewrite <- Rinv_r_sym.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hlt:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H12:- (delta / 2) <=
Rmax (- (delta / 2)) ((a - c) / 2)
H13:delta / 2 >=
- Rmax (- (delta / 2)) ((a - c) / 2)
H14:- Rmax (- (delta / 2)) ((a - c) / 2) <=
delta / 2
1 * delta < 2 * delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hlt:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H12:- (delta / 2) <=
Rmax (- (delta / 2)) ((a - c) / 2)
H13:delta / 2 >=
- Rmax (- (delta / 2)) ((a - c) / 2)
H14:- Rmax (- (delta / 2)) ((a - c) / 2) <=
delta / 2
2 <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hge:Rmax (- (delta / 2)) ((a - c) / 2) >= 0
Rmax (- (delta / 2)) ((a - c) / 2) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  rewrite Rmult_1_l; rewrite double.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hlt:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H12:- (delta / 2) <=
Rmax (- (delta / 2)) ((a - c) / 2)
H13:delta / 2 >=
- Rmax (- (delta / 2)) ((a - c) / 2)
H14:- Rmax (- (delta / 2)) ((a - c) / 2) <=
delta / 2
delta < delta + delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hlt:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H12:- (delta / 2) <=
Rmax (- (delta / 2)) ((a - c) / 2)
H13:delta / 2 >=
- Rmax (- (delta / 2)) ((a - c) / 2)
H14:- Rmax (- (delta / 2)) ((a - c) / 2) <=
delta / 2
2 <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hge:Rmax (- (delta / 2)) ((a - c) / 2) >= 0
Rmax (- (delta / 2)) ((a - c) / 2) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  pattern delta at 2; rewrite <- (Rplus_0_r delta);
    apply Rplus_lt_compat_l; rewrite Rplus_0_r; apply (cond_pos delta).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hlt:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H12:- (delta / 2) <=
Rmax (- (delta / 2)) ((a - c) / 2)
H13:delta / 2 >=
- Rmax (- (delta / 2)) ((a - c) / 2)
H14:- Rmax (- (delta / 2)) ((a - c) / 2) <=
delta / 2
2 <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hge:Rmax (- (delta / 2)) ((a - c) / 2) >= 0
Rmax (- (delta / 2)) ((a - c) / 2) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  discrR.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hge:Rmax (- (delta / 2)) ((a - c) / 2) >= 0
Rmax (- (delta / 2)) ((a - c) / 2) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  cut (- (delta / 2) < 0).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hge:Rmax (- (delta / 2)) ((a - c) / 2) >= 0
- (delta / 2) < 0 ->
Rmax (- (delta / 2)) ((a - c) / 2) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hge:Rmax (- (delta / 2)) ((a - c) / 2) >= 0
- (delta / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  cut ((a - c) / 2 < 0).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hge:Rmax (- (delta / 2)) ((a - c) / 2) >= 0
(a - c) / 2 < 0 ->
- (delta / 2) < 0 ->
Rmax (- (delta / 2)) ((a - c) / 2) < delta
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hge:Rmax (- (delta / 2)) ((a - c) / 2) >= 0
(a - c) / 2 < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hge:Rmax (- (delta / 2)) ((a - c) / 2) >= 0
- (delta / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  intros;
    generalize
      (Rmax_stable_in_negreal (mknegreal (- (delta / 2)) H13)
        (mknegreal ((a - c) / 2) H12)); simpl;
      intro; generalize (Rge_le (Rmax (- (delta / 2)) ((a - c) / 2)) 0 Hge);
        intro;
          elim
            (Rlt_irrefl 0
              (Rle_lt_trans 0 (Rmax (- (delta / 2)) ((a - c) / 2)) 0 H15 H14)).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hge:Rmax (- (delta / 2)) ((a - c) / 2) >= 0
(a - c) / 2 < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hge:Rmax (- (delta / 2)) ((a - c) / 2) >= 0
- (delta / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  rewrite <- Ropp_0; rewrite <- (Ropp_involutive ((a - c) / 2));
    apply Ropp_lt_gt_contravar; replace (- ((a - c) / 2)) with ((c - a) / 2).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hge:Rmax (- (delta / 2)) ((a - c) / 2) >= 0
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hge:Rmax (- (delta / 2)) ((a - c) / 2) >= 0
(c - a) / 2 = - ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hge:Rmax (- (delta / 2)) ((a - c) / 2) >= 0
- (delta / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  assumption.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hge:Rmax (- (delta / 2)) ((a - c) / 2) >= 0
(c - a) / 2 = - ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hge:Rmax (- (delta / 2)) ((a - c) / 2) >= 0
- (delta / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  unfold Rdiv.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hge:Rmax (- (delta / 2)) ((a - c) / 2) >= 0
(c - a) * / 2 = - ((a - c) * / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hge:Rmax (- (delta / 2)) ((a - c) / 2) >= 0
- (delta / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  rewrite <- Ropp_mult_distr_l_reverse.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hge:Rmax (- (delta / 2)) ((a - c) / 2) >= 0
(c - a) * / 2 = - (a - c) * / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hge:Rmax (- (delta / 2)) ((a - c) / 2) >= 0
- (delta / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  rewrite (Ropp_minus_distr a c).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hge:Rmax (- (delta / 2)) ((a - c) / 2) >= 0
(c - a) * / 2 = (c - a) * / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hge:Rmax (- (delta / 2)) ((a - c) / 2) >= 0
- (delta / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  reflexivity.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
H11:Rmax (- (delta / 2)) ((a - c) / 2) <> 0
Hge:Rmax (- (delta / 2)) ((a - c) / 2) >= 0
- (delta / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; unfold Rdiv;
    apply Rmult_lt_0_compat;
      [ apply (cond_pos delta)
        | assert (Hyp : 0 < 2); [ prove_sup0 | apply (Rinv_0_lt_compat 2 Hyp) ] ].f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:Rmax (- (delta / 2)) ((a - c) / 2) < 0
Rmax (- (delta / 2)) ((a - c) / 2) <> 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  red; intro; rewrite H11 in H10; elim (Rlt_irrefl 0 H10).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  cut ((a - c) / 2 < 0).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
(a - c) / 2 < 0 ->
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
(a - c) / 2 < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  intro; cut (- (delta / 2) < 0).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:(a - c) / 2 < 0
- (delta / 2) < 0 ->
Rmax (- (delta / 2)) ((a - c) / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:(a - c) / 2 < 0
- (delta / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
(a - c) / 2 < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  intro;
    apply
      (Rmax_stable_in_negreal (mknegreal (- (delta / 2)) H11)
        (mknegreal ((a - c) / 2) H10)).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
H10:(a - c) / 2 < 0
- (delta / 2) < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
(a - c) / 2 < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; unfold Rdiv;
    apply Rmult_lt_0_compat;
      [ apply (cond_pos delta)
        | assert (Hyp : 0 < 2); [ prove_sup0 | apply (Rinv_0_lt_compat 2 Hyp) ] ].f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
(a - c) / 2 < 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  rewrite <- Ropp_0; rewrite <- (Ropp_involutive ((a - c) / 2));
    apply Ropp_lt_gt_contravar; replace (- ((a - c) / 2)) with ((c - a) / 2).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
(c - a) / 2 = - ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  assumption.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
(c - a) / 2 = - ((a - c) / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  unfold Rdiv.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
(c - a) * / 2 = - ((a - c) * / 2)
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  rewrite <- Ropp_mult_distr_l_reverse.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
(c - a) * / 2 = - (a - c) * / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  rewrite (Ropp_minus_distr a c).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
H8:0 < (c - a) / 2
(c - a) * / 2 = (c - a) * / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  reflexivity.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
H7:0 < - (l / 2)
delta:posreal
H9:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (c + h) - f c) / h - l) < - (l / 2)
0 < (c - a) / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  unfold Rdiv; apply Rmult_lt_0_compat;
    [ generalize (Rplus_lt_compat_r (- a) a c H); rewrite Rplus_opp_r; intro;
      assumption
      | assert (Hyp : 0 < 2); [ prove_sup0 | apply (Rinv_0_lt_compat 2 Hyp) ] ].f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - (l / 2)
  replace (- (l / 2)) with (- l / 2).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - l / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
- l / 2 = - (l / 2)
  unfold Rdiv; apply Rmult_lt_0_compat.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < - l
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
- l / 2 = - (l / 2)
  rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; assumption.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
0 < / 2
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
- l / 2 = - (l / 2)
  assert (Hyp : 0 < 2); [ prove_sup0 | apply (Rinv_0_lt_compat 2 Hyp) ].f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x <= f c
H2:0 = derive_pt f c pr \/ 0 > derive_pt f c pr
l:R
H3:0 > l
H4:exists l0 : R, derive_pt f c pr = l0
H5:derive_pt f c pr = l
H6:derivable_pt_lim f c l
- l / 2 = - (l / 2)
  unfold Rdiv; apply Ropp_mult_distr_l_reverse.
Qed.

Theorem deriv_minimum :
  forall f (a b c:R) (pr:derivable_pt f c),
    a < c ->
    c < b ->
    (forall x:R, a < x -> x < b -> f c <= f x) -> derive_pt f c pr = 0.forall (f : R -> R) (a b c : R)
  (pr : derivable_pt f c),
a < c ->
c < b ->
(forall x : R, a < x -> x < b -> f c <= f x) ->
derive_pt f c pr = 0
Proof.forall (f : R -> R) (a b c : R)
  (pr : derivable_pt f c),
a < c ->
c < b ->
(forall x : R, a < x -> x < b -> f c <= f x) ->
derive_pt f c pr = 0
  intros.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f c <= f x
derive_pt f c pr = 0
  rewrite <- (Ropp_involutive (derive_pt f c pr)).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f c <= f x
- - derive_pt f c pr = 0
  apply Ropp_eq_0_compat.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f c <= f x
- derive_pt f c pr = 0
  rewrite <- (derive_pt_opp f c pr).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f c <= f x
derive_pt (- f) c (derivable_pt_opp f c pr) = 0
  cut (forall x:R, a < x -> x < b -> (- f)%F x <= (- f)%F c).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f c <= f x
(forall x : R,
 a < x -> x < b -> (- f)%F x <= (- f)%F c) ->
derive_pt (- f) c (derivable_pt_opp f c pr) = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f c <= f x
forall x : R, a < x -> x < b -> (- f)%F x <= (- f)%F c
  intro.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f c <= f x
H2:forall x : R,
a < x -> x < b -> (- f)%F x <= (- f)%F c
derive_pt (- f) c (derivable_pt_opp f c pr) = 0
f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f c <= f x
forall x : R, a < x -> x < b -> (- f)%F x <= (- f)%F c
  apply (deriv_maximum (- f)%F a b c (derivable_pt_opp _ _ pr) H H0 H2).f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f c <= f x
forall x : R, a < x -> x < b -> (- f)%F x <= (- f)%F c
  intros; unfold opp_fct; apply Ropp_ge_le_contravar; apply Rle_ge.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x0 : R, a < x0 -> x0 < b -> f c <= f x0
x:R
H2:a < x
H3:x < b
f c <= f x
  apply (H1 x H2 H3).
Qed.

Theorem deriv_constant2 :
  forall f (a b c:R) (pr:derivable_pt f c),
    a < c ->
    c < b -> (forall x:R, a < x -> x < b -> f x = f c) -> derive_pt f c pr = 0.forall (f : R -> R) (a b c : R)
  (pr : derivable_pt f c),
a < c ->
c < b ->
(forall x : R, a < x -> x < b -> f x = f c) ->
derive_pt f c pr = 0
Proof.forall (f : R -> R) (a b c : R)
  (pr : derivable_pt f c),
a < c ->
c < b ->
(forall x : R, a < x -> x < b -> f x = f c) ->
derive_pt f c pr = 0
  intros.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x = f c
derive_pt f c pr = 0
  eapply deriv_maximum with a b; try assumption.f:R -> R
a, b, c:R
pr:derivable_pt f c
H:a < c
H0:c < b
H1:forall x : R, a < x -> x < b -> f x = f c
forall x : R, a < x -> x < b -> f x <= f c
  intros; right; apply (H1 x H2 H3).
Qed.

(**********)
Lemma nonneg_derivative_0 :
  forall f (pr:derivable f),
    increasing f -> forall x:R, 0 <= derive_pt f x (pr x).forall (f : R -> R) (pr : derivable f),
increasing f ->
forall x : R, 0 <= derive_pt f x (pr x)
Proof.forall (f : R -> R) (pr : derivable f),
increasing f ->
forall x : R, 0 <= derive_pt f x (pr x)
  intros; unfold increasing in H.f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
0 <= derive_pt f x (pr x)
  assert (H0 := derivable_derive f x (pr x)).f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l : R, derive_pt f x (pr x) = l
0 <= derive_pt f x (pr x)
  elim H0; intros l H1.f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
0 <= derive_pt f x (pr x)
  rewrite H1; case (Rtotal_order 0 l); intro.f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 < l
0 <= l
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
0 <= l
  left; assumption.f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
0 <= l
  elim H2; intro.f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 = l
0 <= l
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
0 <= l
  right; assumption.f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
0 <= l
  assert (H4 := derive_pt_eq_1 f x l (pr x) H1).f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 <= l
  cut (0 < - (l / 2)).f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2) -> 0 <= l
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  intro; elim (H4 (- (l / 2)) H5); intros delta H6.f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
0 <= l
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  cut (delta / 2 <> 0 /\ 0 < delta / 2 /\ Rabs (delta / 2) < delta).f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta -> 
0 <= l
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  intro; decompose [and] H7; intros; generalize (H6 (delta / 2) H8 H11);
    cut (0 <= (f (x + delta / 2) - f x) / (delta / 2)).f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
0 <= (f (x + delta / 2) - f x) / (delta / 2) ->
Rabs ((f (x + delta / 2) - f x) / (delta / 2) - l) <
- (l / 2) -> 0 <= l
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
0 <= (f (x + delta / 2) - f x) / (delta / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  intro; cut (0 <= (f (x + delta / 2) - f x) / (delta / 2) - l).f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
0 <= (f (x + delta / 2) - f x) / (delta / 2) - l ->
Rabs ((f (x + delta / 2) - f x) / (delta / 2) - l) <
- (l / 2) -> 0 <= l
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
0 <= (f (x + delta / 2) - f x) / (delta / 2) - l
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
0 <= (f (x + delta / 2) - f x) / (delta / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  intro; unfold Rabs;
    case (Rcase_abs ((f (x + delta / 2) - f x) / (delta / 2) - l)) as [Hlt|Hge].f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
H12:0 <= (f (x + delta / 2) - f x) / (delta / 2) - l
Hlt:(f (x + delta / 2) - f x) / (delta / 2) - l < 0
- ((f (x + delta / 2) - f x) / (delta / 2) - l) <
- (l / 2) -> 0 <= l
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
H12:0 <= (f (x + delta / 2) - f x) / (delta / 2) - l
Hge:(f (x + delta / 2) - f x) / (delta / 2) - l >= 0
(f (x + delta / 2) - f x) / (delta / 2) - l <
- (l / 2) -> 0 <= l
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
0 <= (f (x + delta / 2) - f x) / (delta / 2) - l
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
0 <= (f (x + delta / 2) - f x) / (delta / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  elim
      (Rlt_irrefl 0
        (Rle_lt_trans 0 ((f (x + delta / 2) - f x) / (delta / 2) - l) 0 H12 Hlt)).f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
H12:0 <= (f (x + delta / 2) - f x) / (delta / 2) - l
Hge:(f (x + delta / 2) - f x) / (delta / 2) - l >= 0
(f (x + delta / 2) - f x) / (delta / 2) - l <
- (l / 2) -> 0 <= l
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
0 <= (f (x + delta / 2) - f x) / (delta / 2) - l
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
0 <= (f (x + delta / 2) - f x) / (delta / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  intros;
    generalize
      (Rplus_lt_compat_r l ((f (x + delta / 2) - f x) / (delta / 2) - l)
        (- (l / 2)) H13); unfold Rminus;
      replace (- (l / 2) + l) with (l / 2).f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
H12:0 <= (f (x + delta / 2) - f x) / (delta / 2) - l
Hge:(f (x + delta / 2) - f x) / (delta / 2) - l >= 0
H13:(f (x + delta / 2) - f x) / (delta / 2) - l <
- (l / 2)
(f (x + delta / 2) + - f x) / (delta / 2) + - l + l <
l / 2 -> 0 <= l
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
H12:0 <= (f (x + delta / 2) - f x) / (delta / 2) - l
Hge:(f (x + delta / 2) - f x) / (delta / 2) - l >= 0
H13:(f (x + delta / 2) - f x) / (delta / 2) - l <
- (l / 2)
l / 2 = - (l / 2) + l
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
0 <= (f (x + delta / 2) - f x) / (delta / 2) - l
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
0 <= (f (x + delta / 2) - f x) / (delta / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r; intro;
    generalize
      (Rle_lt_trans 0 ((f (x + delta / 2) - f x) / (delta / 2)) (l / 2) H9 H14);
      intro; cut (l / 2 < 0).f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
H12:0 <= (f (x + delta / 2) - f x) / (delta / 2) - l
Hge:(f (x + delta / 2) - f x) / (delta / 2) - l >= 0
H13:(f (x + delta / 2) - f x) / (delta / 2) - l <
- (l / 2)
H14:(f (x + delta / 2) + - f x) / (delta / 2) <
l / 2
H15:0 < l / 2
l / 2 < 0 -> 0 <= l
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
H12:0 <= (f (x + delta / 2) - f x) / (delta / 2) - l
Hge:(f (x + delta / 2) - f x) / (delta / 2) - l >= 0
H13:(f (x + delta / 2) - f x) / (delta / 2) - l <
- (l / 2)
H14:(f (x + delta / 2) + - f x) / (delta / 2) <
l / 2
H15:0 < l / 2
l / 2 < 0
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
H12:0 <= (f (x + delta / 2) - f x) / (delta / 2) - l
Hge:(f (x + delta / 2) - f x) / (delta / 2) - l >= 0
H13:(f (x + delta / 2) - f x) / (delta / 2) - l <
- (l / 2)
l / 2 = - (l / 2) + l
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
0 <= (f (x + delta / 2) - f x) / (delta / 2) - l
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
0 <= (f (x + delta / 2) - f x) / (delta / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  intro; elim (Rlt_irrefl 0 (Rlt_trans 0 (l / 2) 0 H15 H16)).f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
H12:0 <= (f (x + delta / 2) - f x) / (delta / 2) - l
Hge:(f (x + delta / 2) - f x) / (delta / 2) - l >= 0
H13:(f (x + delta / 2) - f x) / (delta / 2) - l <
- (l / 2)
H14:(f (x + delta / 2) + - f x) / (delta / 2) <
l / 2
H15:0 < l / 2
l / 2 < 0
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
H12:0 <= (f (x + delta / 2) - f x) / (delta / 2) - l
Hge:(f (x + delta / 2) - f x) / (delta / 2) - l >= 0
H13:(f (x + delta / 2) - f x) / (delta / 2) - l <
- (l / 2)
l / 2 = - (l / 2) + l
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
0 <= (f (x + delta / 2) - f x) / (delta / 2) - l
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
0 <= (f (x + delta / 2) - f x) / (delta / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  rewrite <- Ropp_0 in H5;
    generalize (Ropp_lt_gt_contravar (-0) (- (l / 2)) H5);
      repeat rewrite Ropp_involutive; intro; assumption.f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
H12:0 <= (f (x + delta / 2) - f x) / (delta / 2) - l
Hge:(f (x + delta / 2) - f x) / (delta / 2) - l >= 0
H13:(f (x + delta / 2) - f x) / (delta / 2) - l <
- (l / 2)
l / 2 = - (l / 2) + l
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
0 <= (f (x + delta / 2) - f x) / (delta / 2) - l
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
0 <= (f (x + delta / 2) - f x) / (delta / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  pattern l at 3; rewrite double_var.f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
H12:0 <= (f (x + delta / 2) - f x) / (delta / 2) - l
Hge:(f (x + delta / 2) - f x) / (delta / 2) - l >= 0
H13:(f (x + delta / 2) - f x) / (delta / 2) - l <
- (l / 2)
l / 2 = - (l / 2) + (l / 2 + l / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
0 <= (f (x + delta / 2) - f x) / (delta / 2) - l
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
0 <= (f (x + delta / 2) - f x) / (delta / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  ring.f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
0 <= (f (x + delta / 2) - f x) / (delta / 2) - l
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
0 <= (f (x + delta / 2) - f x) / (delta / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  unfold Rminus; apply Rplus_le_le_0_compat.f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
0 <= (f (x + delta / 2) + - f x) / (delta / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
0 <= - l
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
0 <= (f (x + delta / 2) - f x) / (delta / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  unfold Rdiv; apply Rmult_le_pos.f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
0 <= f (x + delta * / 2) + - f x
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
0 <= / (delta * / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
0 <= - l
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
0 <= (f (x + delta / 2) - f x) / (delta / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  cut (x <= x + delta * / 2).f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
x <= x + delta * / 2 ->
0 <= f (x + delta * / 2) + - f x
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
x <= x + delta * / 2
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
0 <= / (delta * / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
0 <= - l
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
0 <= (f (x + delta / 2) - f x) / (delta / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  intro; generalize (H x (x + delta * / 2) H12); intro;
    generalize (Rplus_le_compat_l (- f x) (f x) (f (x + delta * / 2)) H13);
      rewrite Rplus_opp_l; rewrite Rplus_comm; intro; assumption.f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
x <= x + delta * / 2
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
0 <= / (delta * / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
0 <= - l
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
0 <= (f (x + delta / 2) - f x) / (delta / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  pattern x at 1; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l;
    left; assumption.f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
0 <= / (delta * / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
0 <= - l
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
0 <= (f (x + delta / 2) - f x) / (delta / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  left; apply Rinv_0_lt_compat; assumption.f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
H9:0 <= (f (x + delta / 2) - f x) / (delta / 2)
0 <= - l
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
0 <= (f (x + delta / 2) - f x) / (delta / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  left; rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; assumption.f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
0 <= (f (x + delta / 2) - f x) / (delta / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  unfold Rdiv; apply Rmult_le_pos.f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
0 <= f (x + delta * / 2) - f x
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
0 <= / (delta * / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  cut (x <= x + delta * / 2).f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
x <= x + delta * / 2 -> 0 <= f (x + delta * / 2) - f x
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
x <= x + delta * / 2
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
0 <= / (delta * / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  intro; generalize (H x (x + delta * / 2) H9); intro;
    generalize (Rplus_le_compat_l (- f x) (f x) (f (x + delta * / 2)) H12);
      rewrite Rplus_opp_l; rewrite Rplus_comm; intro; assumption.f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
x <= x + delta * / 2
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
0 <= / (delta * / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  pattern x at 1; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l;
    left; assumption.f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
H7:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H8:delta / 2 <> 0
H10:0 < delta / 2
H11:Rabs (delta / 2) < delta
0 <= / (delta * / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  left; apply Rinv_0_lt_compat; assumption.f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  split.f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 <> 0
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
0 < delta / 2 /\ Rabs (delta / 2) < delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  unfold Rdiv; apply prod_neq_R0.f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta <> 0
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
/ 2 <> 0
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
0 < delta / 2 /\ Rabs (delta / 2) < delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  generalize (cond_pos delta); intro; red; intro H9; rewrite H9 in H7;
    elim (Rlt_irrefl 0 H7).f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
/ 2 <> 0
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
0 < delta / 2 /\ Rabs (delta / 2) < delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  apply Rinv_neq_0_compat; discrR.f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
0 < delta / 2 /\ Rabs (delta / 2) < delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  split.f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
0 < delta / 2
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
Rabs (delta / 2) < delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  unfold Rdiv; apply Rmult_lt_0_compat;
    [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ].f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
Rabs (delta / 2) < delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  replace (Rabs (delta / 2)) with (delta / 2).f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 < delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 = Rabs (delta / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  unfold Rdiv; apply Rmult_lt_reg_l with 2.f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
0 < 2
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
2 * (delta * / 2) < 2 * delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 = Rabs (delta / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  prove_sup0.f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
2 * (delta * / 2) < 2 * delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 = Rabs (delta / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  rewrite (Rmult_comm 2).f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta * / 2 * 2 < 2 * delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 = Rabs (delta / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ idtac | discrR ].f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta * 1 < 2 * delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 = Rabs (delta / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  rewrite Rmult_1_r.f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta < 2 * delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 = Rabs (delta / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  rewrite double.f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta < delta + delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 = Rabs (delta / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  pattern (pos delta) at 1; rewrite <- Rplus_0_r.f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta + 0 < delta + delta
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 = Rabs (delta / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  apply Rplus_lt_compat_l; apply (cond_pos delta).f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 = Rabs (delta / 2)
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  symmetry ; apply Rabs_right.f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
H5:0 < - (l / 2)
delta:posreal
H6:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < - (l / 2)
delta / 2 >= 0
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  left; change (0 < delta / 2); unfold Rdiv;
    apply Rmult_lt_0_compat;
      [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ].f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - (l / 2)
  unfold Rdiv; rewrite <- Ropp_mult_distr_l_reverse;
    apply Rmult_lt_0_compat.f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < - l
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < / 2
  apply Rplus_lt_reg_l with l.f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
l + 0 < l + - l
f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < / 2
  unfold Rminus; rewrite Rplus_opp_r; rewrite Rplus_0_r; assumption.f:R -> R
pr:derivable f
H:forall x0 y : R, x0 <= y -> f x0 <= f y
x:R
H0:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H1:derive_pt f x (pr x) = l
H2:0 = l \/ 0 > l
H3:0 > l
H4:derivable_pt_lim f x l
0 < / 2
  apply Rinv_0_lt_compat; prove_sup0.
Qed.