(************************************************************************)
(*         *   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 Import Ranalysis1.
Require Import Rtopology.
Local Open Scope R_scope.

(* The Mean Value Theorem *)
Theorem MVT :
  forall (f g:R -> R) (a b:R) (pr1:forall c:R, a < c < b -> derivable_pt f c)
    (pr2:forall c:R, a < c < b -> derivable_pt g c),
    a < b ->
    (forall c:R, a <= c <= b -> continuity_pt f c) ->
    (forall c:R, a <= c <= b -> continuity_pt g c) ->
    exists c : R,
      (exists P : a < c < b,
        (g b - g a) * derive_pt f c (pr1 c P) =
        (f b - f a) * derive_pt g c (pr2 c P)).
forall (f g : R -> R) (a b : R)
  (pr1 : forall c : R, a < c < b -> derivable_pt f c)
  (pr2 : forall c : R, a < c < b -> derivable_pt g c),
a < b ->
(forall c : R, a <= c <= b -> continuity_pt f c) ->
(forall c : R, a <= c <= b -> continuity_pt g c) ->
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
Proof.
forall (f g : R -> R) (a b : R)
  (pr1 : forall c : R, a < c < b -> derivable_pt f c)
  (pr2 : forall c : R, a < c < b -> derivable_pt g c),
a < b ->
(forall c : R, a <= c <= b -> continuity_pt f c) ->
(forall c : R, a <= c <= b -> continuity_pt g c) ->
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
  intros; assert (H2 := Rlt_le _ _ H).
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
  set (h := fun y:R => (g b - g a) * f y - (f b - f a) * g y).
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
  cut (forall c:R, a < c < b -> derivable_pt h c).
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
(forall c : R, a < c < b -> derivable_pt h c) ->
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  intro X; cut (forall c:R, a <= c <= b -> continuity_pt h c).
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
(forall c : R, a <= c <= b -> continuity_pt h c) ->
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  intro; assert (H4 := continuity_ab_maj h a b H2 H3).
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx : R,
  (forall c : R, a <= c <= b -> h c <= h Mx) /\
  a <= Mx <= b
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  assert (H5 := continuity_ab_min h a b H2 H3).
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx : R,
  (forall c : R, a <= c <= b -> h c <= h Mx) /\
  a <= Mx <= b
H5:exists mx : R,
  (forall c : R, a <= c <= b -> h mx <= h c) /\
  a <= mx <= b
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  elim H4; intros Mx H6.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx : R,
  (forall c : R, a <= c <= b -> h mx <= h c) /\
  a <= mx <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  elim H5; intros mx H7.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  cut (h a = h b).
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b ->
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  intro; set (M := h Mx); set (m := h mx).
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  cut
    (forall (c:R) (P:a < c < b),
      derive_pt h c (X c P) =
      (g b - g a) * derive_pt f c (pr1 c P) -
      (f b - f a) * derive_pt g c (pr2 c P)).
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
(forall (c : R) (P : a < c < b),
 derive_pt h c (X c P) =
 (g b - g a) * derive_pt f c (pr1 c P) -
 (f b - f a) * derive_pt g c (pr2 c P)) ->
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  intro; case (Req_dec (h a) M); intro.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  case (Req_dec (h a) m); intro.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  cut (forall c:R, a <= c <= b -> h c = M).
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
(forall c : R, a <= c <= b -> h c = M) ->
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
forall c : R, a <= c <= b -> h c = M
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  intro; cut (a < (a + b) / 2 < b).
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
a < (a + b) / 2 < b ->
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
a < (a + b) / 2 < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
forall c : R, a <= c <= b -> h c = M
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
(*** h constant ***)
  intro; exists ((a + b) / 2).
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
H13:a < (a + b) / 2 < b
exists P : a < (a + b) / 2 < b,
  (g b - g a) *
  derive_pt f ((a + b) / 2) (pr1 ((a + b) / 2) P) =
  (f b - f a) *
  derive_pt g ((a + b) / 2) (pr2 ((a + b) / 2) P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
a < (a + b) / 2 < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
forall c : R, a <= c <= b -> h c = M
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  exists H13.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
H13:a < (a + b) / 2 < b
(g b - g a) *
derive_pt f ((a + b) / 2) (pr1 ((a + b) / 2) H13) =
(f b - f a) *
derive_pt g ((a + b) / 2) (pr2 ((a + b) / 2) H13)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
a < (a + b) / 2 < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
forall c : R, a <= c <= b -> h c = M
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  apply Rminus_diag_uniq; rewrite <- H9; apply deriv_constant2 with a b.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
H13:a < (a + b) / 2 < b
a < (a + b) / 2
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
H13:a < (a + b) / 2 < b
(a + b) / 2 < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
H13:a < (a + b) / 2 < b
forall x : R, a < x -> x < b -> h x = h ((a + b) / 2)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
a < (a + b) / 2 < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
forall c : R, a <= c <= b -> h c = M
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  elim H13; intros; assumption.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
H13:a < (a + b) / 2 < b
(a + b) / 2 < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
H13:a < (a + b) / 2 < b
forall x : R, a < x -> x < b -> h x = h ((a + b) / 2)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
a < (a + b) / 2 < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
forall c : R, a <= c <= b -> h c = M
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  elim H13; intros; assumption.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
H13:a < (a + b) / 2 < b
forall x : R, a < x -> x < b -> h x = h ((a + b) / 2)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
a < (a + b) / 2 < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
forall c : R, a <= c <= b -> h c = M
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  intros; rewrite (H12 ((a + b) / 2)).
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
H13:a < (a + b) / 2 < b
x:R
H14:a < x
H15:x < b
h x = M
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
H13:a < (a + b) / 2 < b
x:R
H14:a < x
H15:x < b
a <= (a + b) / 2 <= b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
a < (a + b) / 2 < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
forall c : R, a <= c <= b -> h c = M
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  apply H12; split; left; assumption.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
H13:a < (a + b) / 2 < b
x:R
H14:a < x
H15:x < b
a <= (a + b) / 2 <= b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
a < (a + b) / 2 < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
forall c : R, a <= c <= b -> h c = M
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  elim H13; intros; split; left; assumption.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
a < (a + b) / 2 < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
forall c : R, a <= c <= b -> h c = M
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  split.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
a < (a + b) / 2
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
(a + b) / 2 < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
forall c : R, a <= c <= b -> h c = M
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  apply Rmult_lt_reg_l with 2.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
0 < 2
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
2 * a < 2 * ((a + b) / 2)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
(a + b) / 2 < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
forall c : R, a <= c <= b -> h c = M
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  prove_sup0.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
2 * a < 2 * ((a + b) / 2)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
(a + b) / 2 < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
forall c : R, a <= c <= b -> h c = M
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
    rewrite <- Rinv_r_sym.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
2 * a < 1 * (a + b)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
2 <> 0
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
(a + b) / 2 < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
forall c : R, a <= c <= b -> h c = M
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; apply H.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
2 <> 0
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
(a + b) / 2 < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
forall c : R, a <= c <= b -> h c = M
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  discrR.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
(a + b) / 2 < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
forall c : R, a <= c <= b -> h c = M
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  apply Rmult_lt_reg_l with 2.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
0 < 2
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
2 * ((a + b) / 2) < 2 * b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
forall c : R, a <= c <= b -> h c = M
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  prove_sup0.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
2 * ((a + b) / 2) < 2 * b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
forall c : R, a <= c <= b -> h c = M
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
    rewrite <- Rinv_r_sym.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
1 * (a + b) < 2 * b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
2 <> 0
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
forall c : R, a <= c <= b -> h c = M
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  rewrite Rmult_1_l; rewrite Rplus_comm; rewrite double;
    apply Rplus_lt_compat_l; apply H.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
H12:forall c : R, a <= c <= b -> h c = M
2 <> 0
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
forall c : R, a <= c <= b -> h c = M
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  discrR.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a = m
forall c : R, a <= c <= b -> h c = M
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  intros; elim H6; intros H13 _.
f, g:R -> R
a, b:R
pr1:forall c0 : R, a < c0 < b -> derivable_pt f c0
pr2:forall c0 : R, a < c0 < b -> derivable_pt g c0
H:a < b
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R, a <= c0 <= b -> continuity_pt g c0
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c0 : R, a < c0 < b -> derivable_pt h c0
H3:forall c0 : R, a <= c0 <= b -> continuity_pt h c0
H4:exists Mx0 : R,
  (forall c0 : R, a <= c0 <= b -> h c0 <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c0 : R, a <= c0 <= b -> h mx0 <= h c0) /\
  a <= mx0 <= b
Mx:R
H6:(forall c0 : R, a <= c0 <= b -> h c0 <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c0 : R, a <= c0 <= b -> h mx <= h c0) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c0 : R) (P : a < c0 < b),
derive_pt h c0 (X c0 P) =
(g b - g a) * derive_pt f c0 (pr1 c0 P) -
(f b - f a) * derive_pt g c0 (pr2 c0 P)
H10:h a = M
H11:h a = m
c:R
H12:a <= c <= b
H13:forall c0 : R, a <= c0 <= b -> h c0 <= h Mx
h c = M
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  elim H7; intros H14 _.
f, g:R -> R
a, b:R
pr1:forall c0 : R, a < c0 < b -> derivable_pt f c0
pr2:forall c0 : R, a < c0 < b -> derivable_pt g c0
H:a < b
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R, a <= c0 <= b -> continuity_pt g c0
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c0 : R, a < c0 < b -> derivable_pt h c0
H3:forall c0 : R, a <= c0 <= b -> continuity_pt h c0
H4:exists Mx0 : R,
  (forall c0 : R, a <= c0 <= b -> h c0 <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c0 : R, a <= c0 <= b -> h mx0 <= h c0) /\
  a <= mx0 <= b
Mx:R
H6:(forall c0 : R, a <= c0 <= b -> h c0 <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c0 : R, a <= c0 <= b -> h mx <= h c0) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c0 : R) (P : a < c0 < b),
derive_pt h c0 (X c0 P) =
(g b - g a) * derive_pt f c0 (pr1 c0 P) -
(f b - f a) * derive_pt g c0 (pr2 c0 P)
H10:h a = M
H11:h a = m
c:R
H12:a <= c <= b
H13:forall c0 : R, a <= c0 <= b -> h c0 <= h Mx
H14:forall c0 : R, a <= c0 <= b -> h mx <= h c0
h c = M
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  apply Rle_antisym.
f, g:R -> R
a, b:R
pr1:forall c0 : R, a < c0 < b -> derivable_pt f c0
pr2:forall c0 : R, a < c0 < b -> derivable_pt g c0
H:a < b
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R, a <= c0 <= b -> continuity_pt g c0
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c0 : R, a < c0 < b -> derivable_pt h c0
H3:forall c0 : R, a <= c0 <= b -> continuity_pt h c0
H4:exists Mx0 : R,
  (forall c0 : R, a <= c0 <= b -> h c0 <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c0 : R, a <= c0 <= b -> h mx0 <= h c0) /\
  a <= mx0 <= b
Mx:R
H6:(forall c0 : R, a <= c0 <= b -> h c0 <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c0 : R, a <= c0 <= b -> h mx <= h c0) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c0 : R) (P : a < c0 < b),
derive_pt h c0 (X c0 P) =
(g b - g a) * derive_pt f c0 (pr1 c0 P) -
(f b - f a) * derive_pt g c0 (pr2 c0 P)
H10:h a = M
H11:h a = m
c:R
H12:a <= c <= b
H13:forall c0 : R, a <= c0 <= b -> h c0 <= h Mx
H14:forall c0 : R, a <= c0 <= b -> h mx <= h c0
h c <= M
f, g:R -> R
a, b:R
pr1:forall c0 : R, a < c0 < b -> derivable_pt f c0
pr2:forall c0 : R, a < c0 < b -> derivable_pt g c0
H:a < b
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R, a <= c0 <= b -> continuity_pt g c0
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c0 : R, a < c0 < b -> derivable_pt h c0
H3:forall c0 : R, a <= c0 <= b -> continuity_pt h c0
H4:exists Mx0 : R,
  (forall c0 : R, a <= c0 <= b -> h c0 <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c0 : R, a <= c0 <= b -> h mx0 <= h c0) /\
  a <= mx0 <= b
Mx:R
H6:(forall c0 : R, a <= c0 <= b -> h c0 <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c0 : R, a <= c0 <= b -> h mx <= h c0) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c0 : R) (P : a < c0 < b),
derive_pt h c0 (X c0 P) =
(g b - g a) * derive_pt f c0 (pr1 c0 P) -
(f b - f a) * derive_pt g c0 (pr2 c0 P)
H10:h a = M
H11:h a = m
c:R
H12:a <= c <= b
H13:forall c0 : R, a <= c0 <= b -> h c0 <= h Mx
H14:forall c0 : R, a <= c0 <= b -> h mx <= h c0
M <= h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  apply H13; apply H12.
f, g:R -> R
a, b:R
pr1:forall c0 : R, a < c0 < b -> derivable_pt f c0
pr2:forall c0 : R, a < c0 < b -> derivable_pt g c0
H:a < b
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R, a <= c0 <= b -> continuity_pt g c0
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c0 : R, a < c0 < b -> derivable_pt h c0
H3:forall c0 : R, a <= c0 <= b -> continuity_pt h c0
H4:exists Mx0 : R,
  (forall c0 : R, a <= c0 <= b -> h c0 <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c0 : R, a <= c0 <= b -> h mx0 <= h c0) /\
  a <= mx0 <= b
Mx:R
H6:(forall c0 : R, a <= c0 <= b -> h c0 <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c0 : R, a <= c0 <= b -> h mx <= h c0) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c0 : R) (P : a < c0 < b),
derive_pt h c0 (X c0 P) =
(g b - g a) * derive_pt f c0 (pr1 c0 P) -
(f b - f a) * derive_pt g c0 (pr2 c0 P)
H10:h a = M
H11:h a = m
c:R
H12:a <= c <= b
H13:forall c0 : R, a <= c0 <= b -> h c0 <= h Mx
H14:forall c0 : R, a <= c0 <= b -> h mx <= h c0
M <= h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  rewrite H10 in H11; rewrite H11; apply H14; apply H12.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  cut (a < mx < b).
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
a < mx < b ->
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
a < mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
(*** h admet un minimum global sur [a,b] ***)
  intro; exists mx.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
H12:a < mx < b
exists P : a < mx < b,
  (g b - g a) * derive_pt f mx (pr1 mx P) =
  (f b - f a) * derive_pt g mx (pr2 mx P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
a < mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  exists H12.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
H12:a < mx < b
(g b - g a) * derive_pt f mx (pr1 mx H12) =
(f b - f a) * derive_pt g mx (pr2 mx H12)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
a < mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  apply Rminus_diag_uniq; rewrite <- H9; apply deriv_minimum with a b.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
H12:a < mx < b
a < mx
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
H12:a < mx < b
mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
H12:a < mx < b
forall x : R, a < x -> x < b -> h mx <= h x
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
a < mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  elim H12; intros; assumption.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
H12:a < mx < b
mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
H12:a < mx < b
forall x : R, a < x -> x < b -> h mx <= h x
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
a < mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  elim H12; intros; assumption.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
H12:a < mx < b
forall x : R, a < x -> x < b -> h mx <= h x
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
a < mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  intros; elim H7; intros.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
H12:a < mx < b
x:R
H13:a < x
H14:x < b
H15:forall c : R, a <= c <= b -> h mx <= h c
H16:a <= mx <= b
h mx <= h x
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
a < mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  apply H15; split; left; assumption.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
a < mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  elim H7; intros _ H12; elim H12; intros; split.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
H12:a <= mx <= b
H13:a <= mx
H14:mx <= b
a < mx
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
H12:a <= mx <= b
H13:a <= mx
H14:mx <= b
mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  inversion H13.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
H12:a <= mx <= b
H13:a <= mx
H14:mx <= b
H15:a < mx
a < mx
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
H12:a <= mx <= b
H13:a <= mx
H14:mx <= b
H15:a = mx
a < mx
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
H12:a <= mx <= b
H13:a <= mx
H14:mx <= b
mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  apply H15.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
H12:a <= mx <= b
H13:a <= mx
H14:mx <= b
H15:a = mx
a < mx
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
H12:a <= mx <= b
H13:a <= mx
H14:mx <= b
mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  rewrite H15 in H11; elim H11; reflexivity.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
H12:a <= mx <= b
H13:a <= mx
H14:mx <= b
mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  inversion H14.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
H12:a <= mx <= b
H13:a <= mx
H14:mx <= b
H15:mx < b
mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
H12:a <= mx <= b
H13:a <= mx
H14:mx <= b
H15:mx = b
mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  apply H15.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a = M
H11:h a <> m
H12:a <= mx <= b
H13:a <= mx
H14:mx <= b
H15:mx = b
mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  rewrite H8 in H11; rewrite <- H15 in H11; elim H11; reflexivity.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  cut (a < Mx < b).
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
a < Mx < b ->
exists (c : R) (P : a < c < b),
  (g b - g a) * derive_pt f c (pr1 c P) =
  (f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
a < Mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
(*** h admet un maximum global sur [a,b] ***)
  intro; exists Mx.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
H11:a < Mx < b
exists P : a < Mx < b,
  (g b - g a) * derive_pt f Mx (pr1 Mx P) =
  (f b - f a) * derive_pt g Mx (pr2 Mx P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
a < Mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  exists H11.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
H11:a < Mx < b
(g b - g a) * derive_pt f Mx (pr1 Mx H11) =
(f b - f a) * derive_pt g Mx (pr2 Mx H11)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
a < Mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  apply Rminus_diag_uniq; rewrite <- H9; apply deriv_maximum with a b.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
H11:a < Mx < b
a < Mx
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
H11:a < Mx < b
Mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
H11:a < Mx < b
forall x : R, a < x -> x < b -> h x <= h Mx
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
a < Mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  elim H11; intros; assumption.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
H11:a < Mx < b
Mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
H11:a < Mx < b
forall x : R, a < x -> x < b -> h x <= h Mx
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
a < Mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  elim H11; intros; assumption.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
H11:a < Mx < b
forall x : R, a < x -> x < b -> h x <= h Mx
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
a < Mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  intros; elim H6; intros; apply H14.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
H11:a < Mx < b
x:R
H12:a < x
H13:x < b
H14:forall c : R, a <= c <= b -> h c <= h Mx
H15:a <= Mx <= b
a <= x <= b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
a < Mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  split; left; assumption.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
a < Mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  elim H6; intros _ H11; elim H11; intros; split.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
H11:a <= Mx <= b
H12:a <= Mx
H13:Mx <= b
a < Mx
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
H11:a <= Mx <= b
H12:a <= Mx
H13:Mx <= b
Mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  inversion H12.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
H11:a <= Mx <= b
H12:a <= Mx
H13:Mx <= b
H14:a < Mx
a < Mx
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
H11:a <= Mx <= b
H12:a <= Mx
H13:Mx <= b
H14:a = Mx
a < Mx
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
H11:a <= Mx <= b
H12:a <= Mx
H13:Mx <= b
Mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  apply H14.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
H11:a <= Mx <= b
H12:a <= Mx
H13:Mx <= b
H14:a = Mx
a < Mx
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
H11:a <= Mx <= b
H12:a <= Mx
H13:Mx <= b
Mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  rewrite H14 in H10; elim H10; reflexivity.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
H11:a <= Mx <= b
H12:a <= Mx
H13:Mx <= b
Mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  inversion H13.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
H11:a <= Mx <= b
H12:a <= Mx
H13:Mx <= b
H14:Mx < b
Mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
H11:a <= Mx <= b
H12:a <= Mx
H13:Mx <= b
H14:Mx = b
Mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  apply H14.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
H9:forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
H10:h a <> M
H11:a <= Mx <= b
H12:a <= Mx
H13:Mx <= b
H14:Mx = b
Mx < b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  rewrite H8 in H10; rewrite <- H14 in H10; elim H10; reflexivity.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
forall (c : R) (P : a < c < b),
derive_pt h c (X c P) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  intros; unfold h;
    replace
    (derive_pt (fun y:R => (g b - g a) * f y - (f b - f a) * g y) c (X c P))
    with
      (derive_pt ((fct_cte (g b - g a) * f)%F - (fct_cte (f b - f a) * g)%F) c
        (derivable_pt_minus _ _ _
          (derivable_pt_mult _ _ _ (derivable_pt_const (g b - g a) c) (pr1 c P))
          (derivable_pt_mult _ _ _ (derivable_pt_const (f b - f a) c) (pr2 c P))));
      [ idtac | apply pr_nu ].
f, g:R -> R
a, b:R
pr1:forall c0 : R, a < c0 < b -> derivable_pt f c0
pr2:forall c0 : R, a < c0 < b -> derivable_pt g c0
H:a < b
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R, a <= c0 <= b -> continuity_pt g c0
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c0 : R, a < c0 < b -> derivable_pt h c0
H3:forall c0 : R, a <= c0 <= b -> continuity_pt h c0
H4:exists Mx0 : R,
  (forall c0 : R, a <= c0 <= b -> h c0 <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c0 : R, a <= c0 <= b -> h mx0 <= h c0) /\
  a <= mx0 <= b
Mx:R
H6:(forall c0 : R, a <= c0 <= b -> h c0 <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c0 : R, a <= c0 <= b -> h mx <= h c0) /\
a <= mx <= b
H8:h a = h b
M:=h Mx:R
m:=h mx:R
c:R
P:a < c < b
derive_pt
  (fct_cte (g b - g a) * f - fct_cte (f b - f a) * g)
  c
  (derivable_pt_minus (fct_cte (g b - g a) * f)%F
     (fct_cte (f b - f a) * g)%F c
     (derivable_pt_mult (fct_cte (g b - g a)) f c
        (derivable_pt_const (g b - g a) c) (pr1 c P))
     (derivable_pt_mult (fct_cte (f b - f a)) g c
        (derivable_pt_const (f b - f a) c) (pr2 c P))) =
(g b - g a) * derive_pt f c (pr1 c P) -
(f b - f a) * derive_pt g c (pr2 c P)
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  rewrite derive_pt_minus; do 2 rewrite derive_pt_mult;
    do 2 rewrite derive_pt_const; do 2 rewrite Rmult_0_l;
      do 2 rewrite Rplus_0_l; reflexivity.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
H3:forall c : R, a <= c <= b -> continuity_pt h c
H4:exists Mx0 : R,
  (forall c : R, a <= c <= b -> h c <= h Mx0) /\
  a <= Mx0 <= b
H5:exists mx0 : R,
  (forall c : R, a <= c <= b -> h mx0 <= h c) /\
  a <= mx0 <= b
Mx:R
H6:(forall c : R, a <= c <= b -> h c <= h Mx) /\
a <= Mx <= b
mx:R
H7:(forall c : R, a <= c <= b -> h mx <= h c) /\
a <= mx <= b
h a = h b
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  unfold h; ring.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c : R, a < c < b -> derivable_pt h c
forall c : R, a <= c <= b -> continuity_pt h c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  intros; unfold h;
    change
      (continuity_pt ((fct_cte (g b - g a) * f)%F - (fct_cte (f b - f a) * g)%F)
        c).
f, g:R -> R
a, b:R
pr1:forall c0 : R, a < c0 < b -> derivable_pt f c0
pr2:forall c0 : R, a < c0 < b -> derivable_pt g c0
H:a < b
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R, a <= c0 <= b -> continuity_pt g c0
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c0 : R, a < c0 < b -> derivable_pt h c0
c:R
H3:a <= c <= b
continuity_pt
  (fct_cte (g b - g a) * f - fct_cte (f b - f a) * g)
  c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  apply continuity_pt_minus; apply continuity_pt_mult.
f, g:R -> R
a, b:R
pr1:forall c0 : R, a < c0 < b -> derivable_pt f c0
pr2:forall c0 : R, a < c0 < b -> derivable_pt g c0
H:a < b
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R, a <= c0 <= b -> continuity_pt g c0
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c0 : R, a < c0 < b -> derivable_pt h c0
c:R
H3:a <= c <= b
continuity_pt (fct_cte (g b - g a)) c
f, g:R -> R
a, b:R
pr1:forall c0 : R, a < c0 < b -> derivable_pt f c0
pr2:forall c0 : R, a < c0 < b -> derivable_pt g c0
H:a < b
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R, a <= c0 <= b -> continuity_pt g c0
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c0 : R, a < c0 < b -> derivable_pt h c0
c:R
H3:a <= c <= b
continuity_pt f c
f, g:R -> R
a, b:R
pr1:forall c0 : R, a < c0 < b -> derivable_pt f c0
pr2:forall c0 : R, a < c0 < b -> derivable_pt g c0
H:a < b
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R, a <= c0 <= b -> continuity_pt g c0
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c0 : R, a < c0 < b -> derivable_pt h c0
c:R
H3:a <= c <= b
continuity_pt (fct_cte (f b - f a)) c
f, g:R -> R
a, b:R
pr1:forall c0 : R, a < c0 < b -> derivable_pt f c0
pr2:forall c0 : R, a < c0 < b -> derivable_pt g c0
H:a < b
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R, a <= c0 <= b -> continuity_pt g c0
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c0 : R, a < c0 < b -> derivable_pt h c0
c:R
H3:a <= c <= b
continuity_pt g c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  apply derivable_continuous_pt; apply derivable_const.
f, g:R -> R
a, b:R
pr1:forall c0 : R, a < c0 < b -> derivable_pt f c0
pr2:forall c0 : R, a < c0 < b -> derivable_pt g c0
H:a < b
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R, a <= c0 <= b -> continuity_pt g c0
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c0 : R, a < c0 < b -> derivable_pt h c0
c:R
H3:a <= c <= b
continuity_pt f c
f, g:R -> R
a, b:R
pr1:forall c0 : R, a < c0 < b -> derivable_pt f c0
pr2:forall c0 : R, a < c0 < b -> derivable_pt g c0
H:a < b
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R, a <= c0 <= b -> continuity_pt g c0
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c0 : R, a < c0 < b -> derivable_pt h c0
c:R
H3:a <= c <= b
continuity_pt (fct_cte (f b - f a)) c
f, g:R -> R
a, b:R
pr1:forall c0 : R, a < c0 < b -> derivable_pt f c0
pr2:forall c0 : R, a < c0 < b -> derivable_pt g c0
H:a < b
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R, a <= c0 <= b -> continuity_pt g c0
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c0 : R, a < c0 < b -> derivable_pt h c0
c:R
H3:a <= c <= b
continuity_pt g c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  apply H0; apply H3.
f, g:R -> R
a, b:R
pr1:forall c0 : R, a < c0 < b -> derivable_pt f c0
pr2:forall c0 : R, a < c0 < b -> derivable_pt g c0
H:a < b
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R, a <= c0 <= b -> continuity_pt g c0
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c0 : R, a < c0 < b -> derivable_pt h c0
c:R
H3:a <= c <= b
continuity_pt (fct_cte (f b - f a)) c
f, g:R -> R
a, b:R
pr1:forall c0 : R, a < c0 < b -> derivable_pt f c0
pr2:forall c0 : R, a < c0 < b -> derivable_pt g c0
H:a < b
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R, a <= c0 <= b -> continuity_pt g c0
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c0 : R, a < c0 < b -> derivable_pt h c0
c:R
H3:a <= c <= b
continuity_pt g c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  apply derivable_continuous_pt; apply derivable_const.
f, g:R -> R
a, b:R
pr1:forall c0 : R, a < c0 < b -> derivable_pt f c0
pr2:forall c0 : R, a < c0 < b -> derivable_pt g c0
H:a < b
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R, a <= c0 <= b -> continuity_pt g c0
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
X:forall c0 : R, a < c0 < b -> derivable_pt h c0
c:R
H3:a <= c <= b
continuity_pt g c
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  apply H1; apply H3.
f, g:R -> R
a, b:R
pr1:forall c : R, a < c < b -> derivable_pt f c
pr2:forall c : R, a < c < b -> derivable_pt g c
H:a < b
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt g c
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
forall c : R, a < c < b -> derivable_pt h c
  intros;
    change
      (derivable_pt ((fct_cte (g b - g a) * f)%F - (fct_cte (f b - f a) * g)%F)
        c).
f, g:R -> R
a, b:R
pr1:forall c0 : R, a < c0 < b -> derivable_pt f c0
pr2:forall c0 : R, a < c0 < b -> derivable_pt g c0
H:a < b
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R, a <= c0 <= b -> continuity_pt g c0
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
c:R
H3:a < c < b
derivable_pt
  (fct_cte (g b - g a) * f - fct_cte (f b - f a) * g)
  c
  apply derivable_pt_minus; apply derivable_pt_mult.
f, g:R -> R
a, b:R
pr1:forall c0 : R, a < c0 < b -> derivable_pt f c0
pr2:forall c0 : R, a < c0 < b -> derivable_pt g c0
H:a < b
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R, a <= c0 <= b -> continuity_pt g c0
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
c:R
H3:a < c < b
derivable_pt (fct_cte (g b - g a)) c
f, g:R -> R
a, b:R
pr1:forall c0 : R, a < c0 < b -> derivable_pt f c0
pr2:forall c0 : R, a < c0 < b -> derivable_pt g c0
H:a < b
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R, a <= c0 <= b -> continuity_pt g c0
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
c:R
H3:a < c < b
derivable_pt f c
f, g:R -> R
a, b:R
pr1:forall c0 : R, a < c0 < b -> derivable_pt f c0
pr2:forall c0 : R, a < c0 < b -> derivable_pt g c0
H:a < b
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R, a <= c0 <= b -> continuity_pt g c0
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
c:R
H3:a < c < b
derivable_pt (fct_cte (f b - f a)) c
f, g:R -> R
a, b:R
pr1:forall c0 : R, a < c0 < b -> derivable_pt f c0
pr2:forall c0 : R, a < c0 < b -> derivable_pt g c0
H:a < b
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R, a <= c0 <= b -> continuity_pt g c0
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
c:R
H3:a < c < b
derivable_pt g c
  apply derivable_pt_const.
f, g:R -> R
a, b:R
pr1:forall c0 : R, a < c0 < b -> derivable_pt f c0
pr2:forall c0 : R, a < c0 < b -> derivable_pt g c0
H:a < b
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R, a <= c0 <= b -> continuity_pt g c0
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
c:R
H3:a < c < b
derivable_pt f c
f, g:R -> R
a, b:R
pr1:forall c0 : R, a < c0 < b -> derivable_pt f c0
pr2:forall c0 : R, a < c0 < b -> derivable_pt g c0
H:a < b
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R, a <= c0 <= b -> continuity_pt g c0
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
c:R
H3:a < c < b
derivable_pt (fct_cte (f b - f a)) c
f, g:R -> R
a, b:R
pr1:forall c0 : R, a < c0 < b -> derivable_pt f c0
pr2:forall c0 : R, a < c0 < b -> derivable_pt g c0
H:a < b
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R, a <= c0 <= b -> continuity_pt g c0
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
c:R
H3:a < c < b
derivable_pt g c
  apply (pr1 _ H3).
f, g:R -> R
a, b:R
pr1:forall c0 : R, a < c0 < b -> derivable_pt f c0
pr2:forall c0 : R, a < c0 < b -> derivable_pt g c0
H:a < b
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R, a <= c0 <= b -> continuity_pt g c0
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
c:R
H3:a < c < b
derivable_pt (fct_cte (f b - f a)) c
f, g:R -> R
a, b:R
pr1:forall c0 : R, a < c0 < b -> derivable_pt f c0
pr2:forall c0 : R, a < c0 < b -> derivable_pt g c0
H:a < b
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R, a <= c0 <= b -> continuity_pt g c0
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
c:R
H3:a < c < b
derivable_pt g c
  apply derivable_pt_const.
f, g:R -> R
a, b:R
pr1:forall c0 : R, a < c0 < b -> derivable_pt f c0
pr2:forall c0 : R, a < c0 < b -> derivable_pt g c0
H:a < b
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R, a <= c0 <= b -> continuity_pt g c0
H2:a <= b
h:=fun y : R => (g b - g a) * f y - (f b - f a) * g y:R -> R
c:R
H3:a < c < b
derivable_pt g c
  apply (pr2 _ H3).
Qed.

(* Corollaries ... *)
Lemma MVT_cor1 :
  forall (f:R -> R) (a b:R) (pr:derivable f),
    a < b ->
    exists c : R, f b - f a = derive_pt f c (pr c) * (b - a) /\ a < c < b.
forall (f : R -> R) (a b : R) (pr : derivable f),
a < b ->
exists c : R,
  f b - f a = derive_pt f c (pr c) * (b - a) /\
  a < c < b
Proof.
forall (f : R -> R) (a b : R) (pr : derivable f),
a < b ->
exists c : R,
  f b - f a = derive_pt f c (pr c) * (b - a) /\
  a < c < b
  intros f a b pr H; cut (forall c:R, a < c < b -> derivable_pt f c);
    [ intro X | intros; apply pr ].
f:R -> R
a, b:R
pr:derivable f
H:a < b
X:forall c : R, a < c < b -> derivable_pt f c
exists c : R,
  f b - f a = derive_pt f c (pr c) * (b - a) /\
  a < c < b
  cut (forall c:R, a < c < b -> derivable_pt id c);
    [ intro X0 | intros; apply derivable_pt_id ].
f:R -> R
a, b:R
pr:derivable f
H:a < b
X:forall c : R, a < c < b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt id c
exists c : R,
  f b - f a = derive_pt f c (pr c) * (b - a) /\
  a < c < b
  cut (forall c:R, a <= c <= b -> continuity_pt f c);
    [ intro | intros; apply derivable_continuous_pt; apply pr ].
f:R -> R
a, b:R
pr:derivable f
H:a < b
X:forall c : R, a < c < b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt id c
H0:forall c : R, a <= c <= b -> continuity_pt f c
exists c : R,
  f b - f a = derive_pt f c (pr c) * (b - a) /\
  a < c < b
  cut (forall c:R, a <= c <= b -> continuity_pt id c);
    [ intro | intros; apply derivable_continuous_pt; apply derivable_id ].
f:R -> R
a, b:R
pr:derivable f
H:a < b
X:forall c : R, a < c < b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt id c
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt id c
exists c : R,
  f b - f a = derive_pt f c (pr c) * (b - a) /\
  a < c < b
  assert (H2 := MVT f id a b X X0 H H0 H1).
f:R -> R
a, b:R
pr:derivable f
H:a < b
X:forall c : R, a < c < b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt id c
H0:forall c : R, a <= c <= b -> continuity_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt id c
H2:exists (c : R) (P : a < c < b),
  (id b - id a) * derive_pt f c (X c P) =
  (f b - f a) * derive_pt id c (X0 c P)
exists c : R,
  f b - f a = derive_pt f c (pr c) * (b - a) /\
  a < c < b
  destruct H2 as (c & P & H4).
f:R -> R
a, b:R
pr:derivable f
H:a < b
X:forall c0 : R, a < c0 < b -> derivable_pt f c0
X0:forall c0 : R, a < c0 < b -> derivable_pt id c0
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R,
a <= c0 <= b -> continuity_pt id c0
c:R
P:a < c < b
H4:(id b - id a) * derive_pt f c (X c P) =
(f b - f a) * derive_pt id c (X0 c P)
exists c0 : R,
  f b - f a = derive_pt f c0 (pr c0) * (b - a) /\
  a < c0 < b
  exists c; split.
f:R -> R
a, b:R
pr:derivable f
H:a < b
X:forall c0 : R, a < c0 < b -> derivable_pt f c0
X0:forall c0 : R, a < c0 < b -> derivable_pt id c0
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R,
a <= c0 <= b -> continuity_pt id c0
c:R
P:a < c < b
H4:(id b - id a) * derive_pt f c (X c P) =
(f b - f a) * derive_pt id c (X0 c P)
f b - f a = derive_pt f c (pr c) * (b - a)
f:R -> R
a, b:R
pr:derivable f
H:a < b
X:forall c0 : R, a < c0 < b -> derivable_pt f c0
X0:forall c0 : R, a < c0 < b -> derivable_pt id c0
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R,
a <= c0 <= b -> continuity_pt id c0
c:R
P:a < c < b
H4:(id b - id a) * derive_pt f c (X c P) =
(f b - f a) * derive_pt id c (X0 c P)
a < c < b
  cut (derive_pt id c (X0 c P) = derive_pt id c (derivable_pt_id c));
    [ intro H5 | apply pr_nu ].
f:R -> R
a, b:R
pr:derivable f
H:a < b
X:forall c0 : R, a < c0 < b -> derivable_pt f c0
X0:forall c0 : R, a < c0 < b -> derivable_pt id c0
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R,
a <= c0 <= b -> continuity_pt id c0
c:R
P:a < c < b
H4:(id b - id a) * derive_pt f c (X c P) =
(f b - f a) * derive_pt id c (X0 c P)
H5:derive_pt id c (X0 c P) =
derive_pt id c (derivable_pt_id c)
f b - f a = derive_pt f c (pr c) * (b - a)
f:R -> R
a, b:R
pr:derivable f
H:a < b
X:forall c0 : R, a < c0 < b -> derivable_pt f c0
X0:forall c0 : R, a < c0 < b -> derivable_pt id c0
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R,
a <= c0 <= b -> continuity_pt id c0
c:R
P:a < c < b
H4:(id b - id a) * derive_pt f c (X c P) =
(f b - f a) * derive_pt id c (X0 c P)
a < c < b
  rewrite H5 in H4; rewrite (derive_pt_id c) in H4; rewrite Rmult_1_r in H4;
    rewrite <- H4; replace (derive_pt f c (X c P)) with (derive_pt f c (pr c));
      [ idtac | apply pr_nu ]; apply Rmult_comm.
f:R -> R
a, b:R
pr:derivable f
H:a < b
X:forall c0 : R, a < c0 < b -> derivable_pt f c0
X0:forall c0 : R, a < c0 < b -> derivable_pt id c0
H0:forall c0 : R, a <= c0 <= b -> continuity_pt f c0
H1:forall c0 : R,
a <= c0 <= b -> continuity_pt id c0
c:R
P:a < c < b
H4:(id b - id a) * derive_pt f c (X c P) =
(f b - f a) * derive_pt id c (X0 c P)
a < c < b
  apply P.
Qed.

Theorem MVT_cor2 :
  forall (f f':R -> R) (a b:R),
    a < b ->
    (forall c:R, a <= c <= b -> derivable_pt_lim f c (f' c)) ->
    exists c : R, f b - f a = f' c * (b - a) /\ a < c < b.
forall (f f' : R -> R) (a b : R),
a < b ->
(forall c : R,
 a <= c <= b -> derivable_pt_lim f c (f' c)) ->
exists c : R, f b - f a = f' c * (b - a) /\ a < c < b
Proof.
forall (f f' : R -> R) (a b : R),
a < b ->
(forall c : R,
 a <= c <= b -> derivable_pt_lim f c (f' c)) ->
exists c : R, f b - f a = f' c * (b - a) /\ a < c < b
  intros f f' a b H H0; cut (forall c:R, a <= c <= b -> derivable_pt f c).
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
(forall c : R, a <= c <= b -> derivable_pt f c) ->
exists c : R, f b - f a = f' c * (b - a) /\ a < c < b
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
forall c : R, a <= c <= b -> derivable_pt f c
  intro X; cut (forall c:R, a < c < b -> derivable_pt f c).
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
(forall c : R, a < c < b -> derivable_pt f c) ->
exists c : R, f b - f a = f' c * (b - a) /\ a < c < b
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
forall c : R, a < c < b -> derivable_pt f c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
forall c : R, a <= c <= b -> derivable_pt f c
  intro X0; cut (forall c:R, a <= c <= b -> continuity_pt f c).
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
(forall c : R, a <= c <= b -> continuity_pt f c) ->
exists c : R, f b - f a = f' c * (b - a) /\ a < c < b
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
forall c : R, a <= c <= b -> continuity_pt f c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
forall c : R, a < c < b -> derivable_pt f c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
forall c : R, a <= c <= b -> derivable_pt f c
  intro; cut (forall c:R, a <= c <= b -> derivable_pt id c).
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
(forall c : R, a <= c <= b -> derivable_pt id c) ->
exists c : R, f b - f a = f' c * (b - a) /\ a < c < b
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
forall c : R, a <= c <= b -> derivable_pt id c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
forall c : R, a <= c <= b -> continuity_pt f c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
forall c : R, a < c < b -> derivable_pt f c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
forall c : R, a <= c <= b -> derivable_pt f c
  intro X1; cut (forall c:R, a < c < b -> derivable_pt id c).
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
(forall c : R, a < c < b -> derivable_pt id c) ->
exists c : R, f b - f a = f' c * (b - a) /\ a < c < b
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
forall c : R, a < c < b -> derivable_pt id c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
forall c : R, a <= c <= b -> derivable_pt id c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
forall c : R, a <= c <= b -> continuity_pt f c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
forall c : R, a < c < b -> derivable_pt f c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
forall c : R, a <= c <= b -> derivable_pt f c
  intro X2; cut (forall c:R, a <= c <= b -> continuity_pt id c).
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
X2:forall c : R, a < c < b -> derivable_pt id c
(forall c : R, a <= c <= b -> continuity_pt id c) ->
exists c : R, f b - f a = f' c * (b - a) /\ a < c < b
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
X2:forall c : R, a < c < b -> derivable_pt id c
forall c : R, a <= c <= b -> continuity_pt id c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
forall c : R, a < c < b -> derivable_pt id c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
forall c : R, a <= c <= b -> derivable_pt id c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
forall c : R, a <= c <= b -> continuity_pt f c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
forall c : R, a < c < b -> derivable_pt f c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
forall c : R, a <= c <= b -> derivable_pt f c
  intro; elim (MVT f id a b X0 X2 H H1 H2); intros x (P,H3).
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
X2:forall c : R, a < c < b -> derivable_pt id c
H2:forall c : R, a <= c <= b -> continuity_pt id c
x:R
P:a < x < b
H3:(id b - id a) * derive_pt f x (X0 x P) =
(f b - f a) * derive_pt id x (X2 x P)
exists c : R, f b - f a = f' c * (b - a) /\ a < c < b
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
X2:forall c : R, a < c < b -> derivable_pt id c
forall c : R, a <= c <= b -> continuity_pt id c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
forall c : R, a < c < b -> derivable_pt id c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
forall c : R, a <= c <= b -> derivable_pt id c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
forall c : R, a <= c <= b -> continuity_pt f c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
forall c : R, a < c < b -> derivable_pt f c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
forall c : R, a <= c <= b -> derivable_pt f c
  exists x; split.
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
X2:forall c : R, a < c < b -> derivable_pt id c
H2:forall c : R, a <= c <= b -> continuity_pt id c
x:R
P:a < x < b
H3:(id b - id a) * derive_pt f x (X0 x P) =
(f b - f a) * derive_pt id x (X2 x P)
f b - f a = f' x * (b - a)
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
X2:forall c : R, a < c < b -> derivable_pt id c
H2:forall c : R, a <= c <= b -> continuity_pt id c
x:R
P:a < x < b
H3:(id b - id a) * derive_pt f x (X0 x P) =
(f b - f a) * derive_pt id x (X2 x P)
a < x < b
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
X2:forall c : R, a < c < b -> derivable_pt id c
forall c : R, a <= c <= b -> continuity_pt id c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
forall c : R, a < c < b -> derivable_pt id c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
forall c : R, a <= c <= b -> derivable_pt id c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
forall c : R, a <= c <= b -> continuity_pt f c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
forall c : R, a < c < b -> derivable_pt f c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
forall c : R, a <= c <= b -> derivable_pt f c
  cut (derive_pt id x (X2 x P) = 1).
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
X2:forall c : R, a < c < b -> derivable_pt id c
H2:forall c : R, a <= c <= b -> continuity_pt id c
x:R
P:a < x < b
H3:(id b - id a) * derive_pt f x (X0 x P) =
(f b - f a) * derive_pt id x (X2 x P)
derive_pt id x (X2 x P) = 1 ->
f b - f a = f' x * (b - a)
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
X2:forall c : R, a < c < b -> derivable_pt id c
H2:forall c : R, a <= c <= b -> continuity_pt id c
x:R
P:a < x < b
H3:(id b - id a) * derive_pt f x (X0 x P) =
(f b - f a) * derive_pt id x (X2 x P)
derive_pt id x (X2 x P) = 1
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
X2:forall c : R, a < c < b -> derivable_pt id c
H2:forall c : R, a <= c <= b -> continuity_pt id c
x:R
P:a < x < b
H3:(id b - id a) * derive_pt f x (X0 x P) =
(f b - f a) * derive_pt id x (X2 x P)
a < x < b
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
X2:forall c : R, a < c < b -> derivable_pt id c
forall c : R, a <= c <= b -> continuity_pt id c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
forall c : R, a < c < b -> derivable_pt id c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
forall c : R, a <= c <= b -> derivable_pt id c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
forall c : R, a <= c <= b -> continuity_pt f c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
forall c : R, a < c < b -> derivable_pt f c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
forall c : R, a <= c <= b -> derivable_pt f c
  cut (derive_pt f x (X0 x P) = f' x).
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
X2:forall c : R, a < c < b -> derivable_pt id c
H2:forall c : R, a <= c <= b -> continuity_pt id c
x:R
P:a < x < b
H3:(id b - id a) * derive_pt f x (X0 x P) =
(f b - f a) * derive_pt id x (X2 x P)
derive_pt f x (X0 x P) = f' x ->
derive_pt id x (X2 x P) = 1 ->
f b - f a = f' x * (b - a)
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
X2:forall c : R, a < c < b -> derivable_pt id c
H2:forall c : R, a <= c <= b -> continuity_pt id c
x:R
P:a < x < b
H3:(id b - id a) * derive_pt f x (X0 x P) =
(f b - f a) * derive_pt id x (X2 x P)
derive_pt f x (X0 x P) = f' x
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
X2:forall c : R, a < c < b -> derivable_pt id c
H2:forall c : R, a <= c <= b -> continuity_pt id c
x:R
P:a < x < b
H3:(id b - id a) * derive_pt f x (X0 x P) =
(f b - f a) * derive_pt id x (X2 x P)
derive_pt id x (X2 x P) = 1
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
X2:forall c : R, a < c < b -> derivable_pt id c
H2:forall c : R, a <= c <= b -> continuity_pt id c
x:R
P:a < x < b
H3:(id b - id a) * derive_pt f x (X0 x P) =
(f b - f a) * derive_pt id x (X2 x P)
a < x < b
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
X2:forall c : R, a < c < b -> derivable_pt id c
forall c : R, a <= c <= b -> continuity_pt id c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
forall c : R, a < c < b -> derivable_pt id c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
forall c : R, a <= c <= b -> derivable_pt id c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
forall c : R, a <= c <= b -> continuity_pt f c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
forall c : R, a < c < b -> derivable_pt f c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
forall c : R, a <= c <= b -> derivable_pt f c
  intros; rewrite H4 in H3; rewrite H5 in H3; unfold id in H3;
    rewrite Rmult_1_r in H3; rewrite Rmult_comm; symmetry ;
      assumption.
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
X2:forall c : R, a < c < b -> derivable_pt id c
H2:forall c : R, a <= c <= b -> continuity_pt id c
x:R
P:a < x < b
H3:(id b - id a) * derive_pt f x (X0 x P) =
(f b - f a) * derive_pt id x (X2 x P)
derive_pt f x (X0 x P) = f' x
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
X2:forall c : R, a < c < b -> derivable_pt id c
H2:forall c : R, a <= c <= b -> continuity_pt id c
x:R
P:a < x < b
H3:(id b - id a) * derive_pt f x (X0 x P) =
(f b - f a) * derive_pt id x (X2 x P)
derive_pt id x (X2 x P) = 1
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
X2:forall c : R, a < c < b -> derivable_pt id c
H2:forall c : R, a <= c <= b -> continuity_pt id c
x:R
P:a < x < b
H3:(id b - id a) * derive_pt f x (X0 x P) =
(f b - f a) * derive_pt id x (X2 x P)
a < x < b
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
X2:forall c : R, a < c < b -> derivable_pt id c
forall c : R, a <= c <= b -> continuity_pt id c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
forall c : R, a < c < b -> derivable_pt id c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
forall c : R, a <= c <= b -> derivable_pt id c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
forall c : R, a <= c <= b -> continuity_pt f c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
forall c : R, a < c < b -> derivable_pt f c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
forall c : R, a <= c <= b -> derivable_pt f c
  apply derive_pt_eq_0; apply H0; elim P; intros; split; left; assumption.
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
X2:forall c : R, a < c < b -> derivable_pt id c
H2:forall c : R, a <= c <= b -> continuity_pt id c
x:R
P:a < x < b
H3:(id b - id a) * derive_pt f x (X0 x P) =
(f b - f a) * derive_pt id x (X2 x P)
derive_pt id x (X2 x P) = 1
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
X2:forall c : R, a < c < b -> derivable_pt id c
H2:forall c : R, a <= c <= b -> continuity_pt id c
x:R
P:a < x < b
H3:(id b - id a) * derive_pt f x (X0 x P) =
(f b - f a) * derive_pt id x (X2 x P)
a < x < b
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
X2:forall c : R, a < c < b -> derivable_pt id c
forall c : R, a <= c <= b -> continuity_pt id c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
forall c : R, a < c < b -> derivable_pt id c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
forall c : R, a <= c <= b -> derivable_pt id c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
forall c : R, a <= c <= b -> continuity_pt f c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
forall c : R, a < c < b -> derivable_pt f c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
forall c : R, a <= c <= b -> derivable_pt f c
  apply derive_pt_eq_0; apply derivable_pt_lim_id.
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
X2:forall c : R, a < c < b -> derivable_pt id c
H2:forall c : R, a <= c <= b -> continuity_pt id c
x:R
P:a < x < b
H3:(id b - id a) * derive_pt f x (X0 x P) =
(f b - f a) * derive_pt id x (X2 x P)
a < x < b
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
X2:forall c : R, a < c < b -> derivable_pt id c
forall c : R, a <= c <= b -> continuity_pt id c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
forall c : R, a < c < b -> derivable_pt id c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
forall c : R, a <= c <= b -> derivable_pt id c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
forall c : R, a <= c <= b -> continuity_pt f c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
forall c : R, a < c < b -> derivable_pt f c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
forall c : R, a <= c <= b -> derivable_pt f c
  assumption.
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
X2:forall c : R, a < c < b -> derivable_pt id c
forall c : R, a <= c <= b -> continuity_pt id c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
forall c : R, a < c < b -> derivable_pt id c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
forall c : R, a <= c <= b -> derivable_pt id c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
forall c : R, a <= c <= b -> continuity_pt f c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
forall c : R, a < c < b -> derivable_pt f c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
forall c : R, a <= c <= b -> derivable_pt f c
  intros; apply derivable_continuous_pt; apply X1; assumption.
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
X1:forall c : R, a <= c <= b -> derivable_pt id c
forall c : R, a < c < b -> derivable_pt id c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
forall c : R, a <= c <= b -> derivable_pt id c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
forall c : R, a <= c <= b -> continuity_pt f c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
forall c : R, a < c < b -> derivable_pt f c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
forall c : R, a <= c <= b -> derivable_pt f c
  intros; apply derivable_pt_id.
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
H1:forall c : R, a <= c <= b -> continuity_pt f c
forall c : R, a <= c <= b -> derivable_pt id c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
forall c : R, a <= c <= b -> continuity_pt f c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
forall c : R, a < c < b -> derivable_pt f c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
forall c : R, a <= c <= b -> derivable_pt f c
  intros; apply derivable_pt_id.
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
X0:forall c : R, a < c < b -> derivable_pt f c
forall c : R, a <= c <= b -> continuity_pt f c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
forall c : R, a < c < b -> derivable_pt f c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
forall c : R, a <= c <= b -> derivable_pt f c
  intros; apply derivable_continuous_pt; apply X; assumption.
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
X:forall c : R, a <= c <= b -> derivable_pt f c
forall c : R, a < c < b -> derivable_pt f c
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
forall c : R, a <= c <= b -> derivable_pt f c
  intros; elim H1; intros; apply X; split; left; assumption.
f, f':R -> R
a, b:R
H:a < b
H0:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
forall c : R, a <= c <= b -> derivable_pt f c
  intros; unfold derivable_pt; exists (f' c); apply H0;
    apply H1.
Qed.

Lemma MVT_cor3 :
  forall (f f':R -> R) (a b:R),
    a < b ->
    (forall x:R, a <= x -> x <= b -> derivable_pt_lim f x (f' x)) ->
    exists c : R, a <= c /\ c <= b /\ f b = f a + f' c * (b - a).
forall (f f' : R -> R) (a b : R),
a < b ->
(forall x : R,
 a <= x -> x <= b -> derivable_pt_lim f x (f' x)) ->
exists c : R,
  a <= c /\ c <= b /\ f b = f a + f' c * (b - a)
Proof.
forall (f f' : R -> R) (a b : R),
a < b ->
(forall x : R,
 a <= x -> x <= b -> derivable_pt_lim f x (f' x)) ->
exists c : R,
  a <= c /\ c <= b /\ f b = f a + f' c * (b - a)
  intros f f' a b H H0;
    assert (H1 :  exists c : R, f b - f a = f' c * (b - a) /\ a < c < b);
      [ apply MVT_cor2; [ apply H | intros; elim H1; intros; apply (H0 _ H2 H3) ]
        | elim H1; intros; exists x; elim H2; intros; elim H4; intros; split;
          [ left; assumption | split; [ left; assumption | rewrite <- H3; ring ] ] ].
Qed.

Lemma Rolle :
  forall (f:R -> R) (a b:R) (pr:forall x:R, a < x < b -> derivable_pt f x),
    (forall x:R, a <= x <= b -> continuity_pt f x) ->
    a < b ->
    f a = f b ->
    exists c : R, (exists P : a < c < b, derive_pt f c (pr c P) = 0).
forall (f : R -> R) (a b : R)
  (pr : forall x : R, a < x < b -> derivable_pt f x),
(forall x : R, a <= x <= b -> continuity_pt f x) ->
a < b ->
f a = f b ->
exists (c : R) (P : a < c < b),
  derive_pt f c (pr c P) = 0
Proof.
forall (f : R -> R) (a b : R)
  (pr : forall x : R, a < x < b -> derivable_pt f x),
(forall x : R, a <= x <= b -> continuity_pt f x) ->
a < b ->
f a = f b ->
exists (c : R) (P : a < c < b),
  derive_pt f c (pr c P) = 0
  intros; assert (H2 : forall x:R, a < x < b -> derivable_pt id x).
f:R -> R
a, b:R
pr:forall x : R, a < x < b -> derivable_pt f x
H:forall x : R, a <= x <= b -> continuity_pt f x
H0:a < b
H1:f a = f b
forall x : R, a < x < b -> derivable_pt id x
f:R -> R
a, b:R
pr:forall x : R, a < x < b -> derivable_pt f x
H:forall x : R, a <= x <= b -> continuity_pt f x
H0:a < b
H1:f a = f b
H2:forall x : R, a < x < b -> derivable_pt id x
exists (c : R) (P : a < c < b),
  derive_pt f c (pr c P) = 0
  intros; apply derivable_pt_id.
f:R -> R
a, b:R
pr:forall x : R, a < x < b -> derivable_pt f x
H:forall x : R, a <= x <= b -> continuity_pt f x
H0:a < b
H1:f a = f b
H2:forall x : R, a < x < b -> derivable_pt id x
exists (c : R) (P : a < c < b),
  derive_pt f c (pr c P) = 0
  assert (H3 := MVT f id a b pr H2 H0 H);
    assert (H4 : forall x:R, a <= x <= b -> continuity_pt id x).
f:R -> R
a, b:R
pr:forall x : R, a < x < b -> derivable_pt f x
H:forall x : R, a <= x <= b -> continuity_pt f x
H0:a < b
H1:f a = f b
H2:forall x : R, a < x < b -> derivable_pt id x
H3:(forall c : R, a <= c <= b -> continuity_pt id c) ->
exists (c : R) (P : a < c < b),
  (id b - id a) * derive_pt f c (pr c P) =
  (f b - f a) * derive_pt id c (H2 c P)
forall x : R, a <= x <= b -> continuity_pt id x
f:R -> R
a, b:R
pr:forall x : R, a < x < b -> derivable_pt f x
H:forall x : R, a <= x <= b -> continuity_pt f x
H0:a < b
H1:f a = f b
H2:forall x : R, a < x < b -> derivable_pt id x
H3:(forall c : R, a <= c <= b -> continuity_pt id c) ->
exists (c : R) (P : a < c < b),
  (id b - id a) * derive_pt f c (pr c P) =
  (f b - f a) * derive_pt id c (H2 c P)
H4:forall x : R, a <= x <= b -> continuity_pt id x
exists (c : R) (P : a < c < b),
  derive_pt f c (pr c P) = 0
  intros; apply derivable_continuous; apply derivable_id.
f:R -> R
a, b:R
pr:forall x : R, a < x < b -> derivable_pt f x
H:forall x : R, a <= x <= b -> continuity_pt f x
H0:a < b
H1:f a = f b
H2:forall x : R, a < x < b -> derivable_pt id x
H3:(forall c : R, a <= c <= b -> continuity_pt id c) ->
exists (c : R) (P : a < c < b),
  (id b - id a) * derive_pt f c (pr c P) =
  (f b - f a) * derive_pt id c (H2 c P)
H4:forall x : R, a <= x <= b -> continuity_pt id x
exists (c : R) (P : a < c < b),
  derive_pt f c (pr c P) = 0
  destruct (H3 H4) as (c & P & H6).
f:R -> R
a, b:R
pr:forall x : R, a < x < b -> derivable_pt f x
H:forall x : R, a <= x <= b -> continuity_pt f x
H0:a < b
H1:f a = f b
H2:forall x : R, a < x < b -> derivable_pt id x
H3:(forall c0 : R,
 a <= c0 <= b -> continuity_pt id c0) ->
exists (c0 : R) (P0 : a < c0 < b),
  (id b - id a) * derive_pt f c0 (pr c0 P0) =
  (f b - f a) * derive_pt id c0 (H2 c0 P0)
H4:forall x : R, a <= x <= b -> continuity_pt id x
c:R
P:a < c < b
H6:(id b - id a) * derive_pt f c (pr c P) =
(f b - f a) * derive_pt id c (H2 c P)
exists (c0 : R) (P0 : a < c0 < b),
  derive_pt f c0 (pr c0 P0) = 0 exists c; exists P; rewrite H1 in H6.
f:R -> R
a, b:R
pr:forall x : R, a < x < b -> derivable_pt f x
H:forall x : R, a <= x <= b -> continuity_pt f x
H0:a < b
H1:f a = f b
H2:forall x : R, a < x < b -> derivable_pt id x
H3:(forall c0 : R,
 a <= c0 <= b -> continuity_pt id c0) ->
exists (c0 : R) (P0 : a < c0 < b),
  (id b - id a) * derive_pt f c0 (pr c0 P0) =
  (f b - f a) * derive_pt id c0 (H2 c0 P0)
H4:forall x : R, a <= x <= b -> continuity_pt id x
c:R
P:a < c < b
H6:(id b - id a) * derive_pt f c (pr c P) =
(f b - f b) * derive_pt id c (H2 c P)
derive_pt f c (pr c P) = 0
  unfold id in H6; unfold Rminus in H6; rewrite Rplus_opp_r in H6.
f:R -> R
a, b:R
pr:forall x : R, a < x < b -> derivable_pt f x
H:forall x : R, a <= x <= b -> continuity_pt f x
H0:a < b
H1:f a = f b
H2:forall x : R, a < x < b -> derivable_pt id x
H3:(forall c0 : R,
 a <= c0 <= b -> continuity_pt id c0) ->
exists (c0 : R) (P0 : a < c0 < b),
  (id b - id a) * derive_pt f c0 (pr c0 P0) =
  (f b - f a) * derive_pt id c0 (H2 c0 P0)
H4:forall x : R, a <= x <= b -> continuity_pt id x
c:R
P:a < c < b
H6:(b + - a) * derive_pt f c (pr c P) =
0 * derive_pt (fun x : R => x) c (H2 c P)
derive_pt f c (pr c P) = 0
  rewrite Rmult_0_l in H6; apply Rmult_eq_reg_l with (b - a);
    [ rewrite Rmult_0_r; apply H6
      | apply Rminus_eq_contra; red; intro H7; rewrite H7 in H0;
        elim (Rlt_irrefl _ H0) ].
Qed.

(**********)
Lemma nonneg_derivative_1 :
  forall (f:R -> R) (pr:derivable f),
    (forall x:R, 0 <= derive_pt f x (pr x)) -> increasing f.
forall (f : R -> R) (pr : derivable f),
(forall x : R, 0 <= derive_pt f x (pr x)) ->
increasing f
Proof.
forall (f : R -> R) (pr : derivable f),
(forall x : R, 0 <= derive_pt f x (pr x)) ->
increasing f
  intros.
f:R -> R
pr:derivable f
H:forall x : R, 0 <= derive_pt f x (pr x)
increasing f
  unfold increasing.
f:R -> R
pr:derivable f
H:forall x : R, 0 <= derive_pt f x (pr x)
forall x y : R, x <= y -> f x <= f y
  intros.
f:R -> R
pr:derivable f
H:forall x0 : R, 0 <= derive_pt f x0 (pr x0)
x, y:R
H0:x <= y
f x <= f y
  destruct (total_order_T x y) as [[H1| ->]|H1].
f:R -> R
pr:derivable f
H:forall x0 : R, 0 <= derive_pt f x0 (pr x0)
x, y:R
H0:x <= y
H1:x < y
f x <= f y
f:R -> R
pr:derivable f
H:forall x : R, 0 <= derive_pt f x (pr x)
y:R
H0:y <= y
f y <= f y
f:R -> R
pr:derivable f
H:forall x0 : R, 0 <= derive_pt f x0 (pr x0)
x, y:R
H0:x <= y
H1:x > y
f x <= f y
  apply Rplus_le_reg_l with (- f x).
f:R -> R
pr:derivable f
H:forall x0 : R, 0 <= derive_pt f x0 (pr x0)
x, y:R
H0:x <= y
H1:x < y
- f x + f x <= - f x + f y
f:R -> R
pr:derivable f
H:forall x : R, 0 <= derive_pt f x (pr x)
y:R
H0:y <= y
f y <= f y
f:R -> R
pr:derivable f
H:forall x0 : R, 0 <= derive_pt f x0 (pr x0)
x, y:R
H0:x <= y
H1:x > y
f x <= f y
  rewrite Rplus_opp_l; rewrite Rplus_comm.
f:R -> R
pr:derivable f
H:forall x0 : R, 0 <= derive_pt f x0 (pr x0)
x, y:R
H0:x <= y
H1:x < y
0 <= f y + - f x
f:R -> R
pr:derivable f
H:forall x : R, 0 <= derive_pt f x (pr x)
y:R
H0:y <= y
f y <= f y
f:R -> R
pr:derivable f
H:forall x0 : R, 0 <= derive_pt f x0 (pr x0)
x, y:R
H0:x <= y
H1:x > y
f x <= f y
  pose proof (MVT_cor1 f _ _ pr H1) as (c & H3 & H4).
f:R -> R
pr:derivable f
H:forall x0 : R, 0 <= derive_pt f x0 (pr x0)
x, y:R
H0:x <= y
H1:x < y
c:R
H3:f y - f x = derive_pt f c (pr c) * (y - x)
H4:x < c < y
0 <= f y + - f x
f:R -> R
pr:derivable f
H:forall x : R, 0 <= derive_pt f x (pr x)
y:R
H0:y <= y
f y <= f y
f:R -> R
pr:derivable f
H:forall x0 : R, 0 <= derive_pt f x0 (pr x0)
x, y:R
H0:x <= y
H1:x > y
f x <= f y
  unfold Rminus in H3.
f:R -> R
pr:derivable f
H:forall x0 : R, 0 <= derive_pt f x0 (pr x0)
x, y:R
H0:x <= y
H1:x < y
c:R
H3:f y + - f x = derive_pt f c (pr c) * (y + - x)
H4:x < c < y
0 <= f y + - f x
f:R -> R
pr:derivable f
H:forall x : R, 0 <= derive_pt f x (pr x)
y:R
H0:y <= y
f y <= f y
f:R -> R
pr:derivable f
H:forall x0 : R, 0 <= derive_pt f x0 (pr x0)
x, y:R
H0:x <= y
H1:x > y
f x <= f y
  rewrite H3.
f:R -> R
pr:derivable f
H:forall x0 : R, 0 <= derive_pt f x0 (pr x0)
x, y:R
H0:x <= y
H1:x < y
c:R
H3:f y + - f x = derive_pt f c (pr c) * (y + - x)
H4:x < c < y
0 <= derive_pt f c (pr c) * (y + - x)
f:R -> R
pr:derivable f
H:forall x : R, 0 <= derive_pt f x (pr x)
y:R
H0:y <= y
f y <= f y
f:R -> R
pr:derivable f
H:forall x0 : R, 0 <= derive_pt f x0 (pr x0)
x, y:R
H0:x <= y
H1:x > y
f x <= f y
  apply Rmult_le_pos.
f:R -> R
pr:derivable f
H:forall x0 : R, 0 <= derive_pt f x0 (pr x0)
x, y:R
H0:x <= y
H1:x < y
c:R
H3:f y + - f x = derive_pt f c (pr c) * (y + - x)
H4:x < c < y
0 <= derive_pt f c (pr c)
f:R -> R
pr:derivable f
H:forall x0 : R, 0 <= derive_pt f x0 (pr x0)
x, y:R
H0:x <= y
H1:x < y
c:R
H3:f y + - f x = derive_pt f c (pr c) * (y + - x)
H4:x < c < y
0 <= y + - x
f:R -> R
pr:derivable f
H:forall x : R, 0 <= derive_pt f x (pr x)
y:R
H0:y <= y
f y <= f y
f:R -> R
pr:derivable f
H:forall x0 : R, 0 <= derive_pt f x0 (pr x0)
x, y:R
H0:x <= y
H1:x > y
f x <= f y
  apply H.
f:R -> R
pr:derivable f
H:forall x0 : R, 0 <= derive_pt f x0 (pr x0)
x, y:R
H0:x <= y
H1:x < y
c:R
H3:f y + - f x = derive_pt f c (pr c) * (y + - x)
H4:x < c < y
0 <= y + - x
f:R -> R
pr:derivable f
H:forall x : R, 0 <= derive_pt f x (pr x)
y:R
H0:y <= y
f y <= f y
f:R -> R
pr:derivable f
H:forall x0 : R, 0 <= derive_pt f x0 (pr x0)
x, y:R
H0:x <= y
H1:x > y
f x <= f y
  apply Rplus_le_reg_l with x.
f:R -> R
pr:derivable f
H:forall x0 : R, 0 <= derive_pt f x0 (pr x0)
x, y:R
H0:x <= y
H1:x < y
c:R
H3:f y + - f x = derive_pt f c (pr c) * (y + - x)
H4:x < c < y
x + 0 <= x + (y + - x)
f:R -> R
pr:derivable f
H:forall x : R, 0 <= derive_pt f x (pr x)
y:R
H0:y <= y
f y <= f y
f:R -> R
pr:derivable f
H:forall x0 : R, 0 <= derive_pt f x0 (pr x0)
x, y:R
H0:x <= y
H1:x > y
f x <= f y
  rewrite Rplus_0_r; replace (x + (y + - x)) with y; [ assumption | ring ].
f:R -> R
pr:derivable f
H:forall x : R, 0 <= derive_pt f x (pr x)
y:R
H0:y <= y
f y <= f y
f:R -> R
pr:derivable f
H:forall x0 : R, 0 <= derive_pt f x0 (pr x0)
x, y:R
H0:x <= y
H1:x > y
f x <= f y
  right; reflexivity.
f:R -> R
pr:derivable f
H:forall x0 : R, 0 <= derive_pt f x0 (pr x0)
x, y:R
H0:x <= y
H1:x > y
f x <= f y
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H0 H1)).
Qed.

(**********)
Lemma nonpos_derivative_0 :
  forall (f:R -> R) (pr:derivable f),
    decreasing f -> forall x:R, derive_pt f x (pr x) <= 0.
forall (f : R -> R) (pr : derivable f),
decreasing f ->
forall x : R, derive_pt f x (pr x) <= 0
Proof.
forall (f : R -> R) (pr : derivable f),
decreasing f ->
forall x : R, derive_pt f x (pr x) <= 0
  intros f pr H x; assert (H0 := H); unfold decreasing in H0;
    generalize (derivable_derive f x (pr x)); intro; elim H1;
      intros l H2.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
derive_pt f x (pr x) <= 0
  rewrite H2; case (Rtotal_order l 0); intro.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l < 0
l <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
l <= 0
  left; assumption.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
l <= 0
  elim H3; intro.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l = 0
l <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
l <= 0
  right; assumption.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
l <= 0
  generalize (derive_pt_eq_1 f x l (pr x) H2); intros; cut (0 < l / 2).
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2 -> l <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  intro; elim (H5 (l / 2) H6); intros delta H7;
    cut (delta / 2 <> 0 /\ 0 < delta / 2 /\ Rabs (delta / 2) < delta).
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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 -> l <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  intro; decompose [and] H8; intros; generalize (H7 (delta / 2) H9 H12);
    cut ((f (x + delta / 2) - f x) / (delta / 2) <= 0).
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0 ->
Rabs ((f (x + delta / 2) - f x) / (delta / 2) - l) <
l / 2 -> l <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 < - ((f (x + delta / 2) - f x) / (delta / 2) - l) ->
Rabs ((f (x + delta / 2) - f x) / (delta / 2) - l) <
l / 2 -> l <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 < - ((f (x + delta / 2) - f x) / (delta / 2) - l)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((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 -> l <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((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 ->
l <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 < - ((f (x + delta / 2) - f x) / (delta / 2) - l)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5: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) H14); unfold Rminus.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
Hlt:(f (x + delta / 2) - f x) / (delta / 2) - l < 0
H14:- ((f (x + delta / 2) - f x) / (delta / 2) - l) <
l / 2
- ((f (x + delta / 2) + - f x) / (delta / 2) + - l) +
- l < l / 2 + - l -> l <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((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 ->
l <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 < - ((f (x + delta / 2) - f x) / (delta / 2) - l)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  replace (l / 2 + - l) with (- (l / 2)).
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
Hlt:(f (x + delta / 2) - f x) / (delta / 2) - l < 0
H14:- ((f (x + delta / 2) - f x) / (delta / 2) - l) <
l / 2
- ((f (x + delta / 2) + - f x) / (delta / 2) + - l) +
- l < - (l / 2) -> l <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
Hlt:(f (x + delta / 2) - f x) / (delta / 2) - l < 0
H14:- ((f (x + delta / 2) - f x) / (delta / 2) - l) <
l / 2
- (l / 2) = l / 2 + - l
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((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 ->
l <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 < - ((f (x + delta / 2) - f x) / (delta / 2) - l)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  replace (- ((f (x + delta / 2) + - f x) / (delta / 2) + - l) + - l) with
  (- ((f (x + delta / 2) + - f x) / (delta / 2))).
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
Hlt:(f (x + delta / 2) - f x) / (delta / 2) - l < 0
H14:- ((f (x + delta / 2) - f x) / (delta / 2) - l) <
l / 2
- ((f (x + delta / 2) + - f x) / (delta / 2)) <
- (l / 2) -> l <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
Hlt:(f (x + delta / 2) - f x) / (delta / 2) - l < 0
H14:- ((f (x + delta / 2) - f x) / (delta / 2) - l) <
l / 2
- ((f (x + delta / 2) + - f x) / (delta / 2)) =
- ((f (x + delta / 2) + - f x) / (delta / 2) + - l) +
- l
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
Hlt:(f (x + delta / 2) - f x) / (delta / 2) - l < 0
H14:- ((f (x + delta / 2) - f x) / (delta / 2) - l) <
l / 2
- (l / 2) = l / 2 + - l
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((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 ->
l <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 < - ((f (x + delta / 2) - f x) / (delta / 2) - l)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  intro.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
Hlt:(f (x + delta / 2) - f x) / (delta / 2) - l < 0
H14:- ((f (x + delta / 2) - f x) / (delta / 2) - l) <
l / 2
H15:- ((f (x + delta / 2) + - f x) / (delta / 2)) <
- (l / 2)
l <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
Hlt:(f (x + delta / 2) - f x) / (delta / 2) - l < 0
H14:- ((f (x + delta / 2) - f x) / (delta / 2) - l) <
l / 2
- ((f (x + delta / 2) + - f x) / (delta / 2)) =
- ((f (x + delta / 2) + - f x) / (delta / 2) + - l) +
- l
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
Hlt:(f (x + delta / 2) - f x) / (delta / 2) - l < 0
H14:- ((f (x + delta / 2) - f x) / (delta / 2) - l) <
l / 2
- (l / 2) = l / 2 + - l
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((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 ->
l <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 < - ((f (x + delta / 2) - f x) / (delta / 2) - l)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  generalize
    (Ropp_lt_gt_contravar (- ((f (x + delta / 2) + - f x) / (delta / 2)))
      (- (l / 2)) H15).
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
Hlt:(f (x + delta / 2) - f x) / (delta / 2) - l < 0
H14:- ((f (x + delta / 2) - f x) / (delta / 2) - l) <
l / 2
H15:- ((f (x + delta / 2) + - f x) / (delta / 2)) <
- (l / 2)
- - ((f (x + delta / 2) + - f x) / (delta / 2)) >
- - (l / 2) -> l <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
Hlt:(f (x + delta / 2) - f x) / (delta / 2) - l < 0
H14:- ((f (x + delta / 2) - f x) / (delta / 2) - l) <
l / 2
- ((f (x + delta / 2) + - f x) / (delta / 2)) =
- ((f (x + delta / 2) + - f x) / (delta / 2) + - l) +
- l
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
Hlt:(f (x + delta / 2) - f x) / (delta / 2) - l < 0
H14:- ((f (x + delta / 2) - f x) / (delta / 2) - l) <
l / 2
- (l / 2) = l / 2 + - l
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((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 ->
l <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 < - ((f (x + delta / 2) - f x) / (delta / 2) - l)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  repeat rewrite Ropp_involutive.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
Hlt:(f (x + delta / 2) - f x) / (delta / 2) - l < 0
H14:- ((f (x + delta / 2) - f x) / (delta / 2) - l) <
l / 2
H15:- ((f (x + delta / 2) + - f x) / (delta / 2)) <
- (l / 2)
(f (x + delta / 2) + - f x) / (delta / 2) > l / 2 ->
l <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
Hlt:(f (x + delta / 2) - f x) / (delta / 2) - l < 0
H14:- ((f (x + delta / 2) - f x) / (delta / 2) - l) <
l / 2
- ((f (x + delta / 2) + - f x) / (delta / 2)) =
- ((f (x + delta / 2) + - f x) / (delta / 2) + - l) +
- l
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
Hlt:(f (x + delta / 2) - f x) / (delta / 2) - l < 0
H14:- ((f (x + delta / 2) - f x) / (delta / 2) - l) <
l / 2
- (l / 2) = l / 2 + - l
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((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 ->
l <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 < - ((f (x + delta / 2) - f x) / (delta / 2) - l)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  intro.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
Hlt:(f (x + delta / 2) - f x) / (delta / 2) - l < 0
H14:- ((f (x + delta / 2) - f x) / (delta / 2) - l) <
l / 2
H15:- ((f (x + delta / 2) + - f x) / (delta / 2)) <
- (l / 2)
H16:(f (x + delta / 2) + - f x) / (delta / 2) >
l / 2
l <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
Hlt:(f (x + delta / 2) - f x) / (delta / 2) - l < 0
H14:- ((f (x + delta / 2) - f x) / (delta / 2) - l) <
l / 2
- ((f (x + delta / 2) + - f x) / (delta / 2)) =
- ((f (x + delta / 2) + - f x) / (delta / 2) + - l) +
- l
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
Hlt:(f (x + delta / 2) - f x) / (delta / 2) - l < 0
H14:- ((f (x + delta / 2) - f x) / (delta / 2) - l) <
l / 2
- (l / 2) = l / 2 + - l
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((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 ->
l <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 < - ((f (x + delta / 2) - f x) / (delta / 2) - l)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  generalize
    (Rlt_trans 0 (l / 2) ((f (x + delta / 2) - f x) / (delta / 2)) H6 H16);
    intro.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
Hlt:(f (x + delta / 2) - f x) / (delta / 2) - l < 0
H14:- ((f (x + delta / 2) - f x) / (delta / 2) - l) <
l / 2
H15:- ((f (x + delta / 2) + - f x) / (delta / 2)) <
- (l / 2)
H16:(f (x + delta / 2) + - f x) / (delta / 2) >
l / 2
H17:0 < (f (x + delta / 2) - f x) / (delta / 2)
l <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
Hlt:(f (x + delta / 2) - f x) / (delta / 2) - l < 0
H14:- ((f (x + delta / 2) - f x) / (delta / 2) - l) <
l / 2
- ((f (x + delta / 2) + - f x) / (delta / 2)) =
- ((f (x + delta / 2) + - f x) / (delta / 2) + - l) +
- l
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
Hlt:(f (x + delta / 2) - f x) / (delta / 2) - l < 0
H14:- ((f (x + delta / 2) - f x) / (delta / 2) - l) <
l / 2
- (l / 2) = l / 2 + - l
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((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 ->
l <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 < - ((f (x + delta / 2) - f x) / (delta / 2) - l)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  elim
    (Rlt_irrefl 0
      (Rlt_le_trans 0 ((f (x + delta / 2) - f x) / (delta / 2)) 0 H17 H10)).
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
Hlt:(f (x + delta / 2) - f x) / (delta / 2) - l < 0
H14:- ((f (x + delta / 2) - f x) / (delta / 2) - l) <
l / 2
- ((f (x + delta / 2) + - f x) / (delta / 2)) =
- ((f (x + delta / 2) + - f x) / (delta / 2) + - l) +
- l
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
Hlt:(f (x + delta / 2) - f x) / (delta / 2) - l < 0
H14:- ((f (x + delta / 2) - f x) / (delta / 2) - l) <
l / 2
- (l / 2) = l / 2 + - l
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((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 ->
l <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 < - ((f (x + delta / 2) - f x) / (delta / 2) - l)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  ring.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
Hlt:(f (x + delta / 2) - f x) / (delta / 2) - l < 0
H14:- ((f (x + delta / 2) - f x) / (delta / 2) - l) <
l / 2
- (l / 2) = l / 2 + - l
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((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 ->
l <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 < - ((f (x + delta / 2) - f x) / (delta / 2) - l)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  pattern l at 3; rewrite double_var.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
Hlt:(f (x + delta / 2) - f x) / (delta / 2) - l < 0
H14:- ((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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((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 ->
l <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 < - ((f (x + delta / 2) - f x) / (delta / 2) - l)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  ring.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((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 ->
l <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 < - ((f (x + delta / 2) - f x) / (delta / 2) - l)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  intros.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
Hge:(f (x + delta / 2) - f x) / (delta / 2) - l >= 0
H14:(f (x + delta / 2) - f x) / (delta / 2) - l <
l / 2
l <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 < - ((f (x + delta / 2) - f x) / (delta / 2) - l)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  generalize
    (Ropp_ge_le_contravar ((f (x + delta / 2) - f x) / (delta / 2) - l) _ Hge).
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
Hge:(f (x + delta / 2) - f x) / (delta / 2) - l >= 0
H14:(f (x + delta / 2) - f x) / (delta / 2) - l <
l / 2
- ((f (x + delta / 2) - f x) / (delta / 2) - l) <= - 0 ->
l <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 < - ((f (x + delta / 2) - f x) / (delta / 2) - l)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  rewrite Ropp_0.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
Hge:(f (x + delta / 2) - f x) / (delta / 2) - l >= 0
H14:(f (x + delta / 2) - f x) / (delta / 2) - l <
l / 2
- ((f (x + delta / 2) - f x) / (delta / 2) - l) <= 0 ->
l <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 < - ((f (x + delta / 2) - f x) / (delta / 2) - l)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  intro.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
H13:0 <
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
Hge:(f (x + delta / 2) - f x) / (delta / 2) - l >= 0
H14:(f (x + delta / 2) - f x) / (delta / 2) - l <
l / 2
H15:- ((f (x + delta / 2) - f x) / (delta / 2) - l) <=
0
l <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 < - ((f (x + delta / 2) - f x) / (delta / 2) - l)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  elim
    (Rlt_irrefl 0
      (Rlt_le_trans 0 (- ((f (x + delta / 2) - f x) / (delta / 2) - l)) 0 H13
        H15)).
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 < - ((f (x + delta / 2) - f x) / (delta / 2) - l)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  replace (- ((f (x + delta / 2) - f x) / (delta / 2) - l)) with
  ((f x - f (x + delta / 2)) / (delta / 2) + l).
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 < (f x - f (x + delta / 2)) / (delta / 2) + l
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
(f x - f (x + delta / 2)) / (delta / 2) + l =
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  unfold Rminus.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 < (f x + - f (x + delta / 2)) / (delta / 2) + l
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
(f x - f (x + delta / 2)) / (delta / 2) + l =
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  apply Rplus_le_lt_0_compat.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 <= (f x + - f (x + delta / 2)) / (delta / 2)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 < l
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
(f x - f (x + delta / 2)) / (delta / 2) + l =
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  unfold Rdiv; apply Rmult_le_pos.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 <= f x + - f (x + delta * / 2)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 <= / (delta * / 2)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 < l
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
(f x - f (x + delta / 2)) / (delta / 2) + l =
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  cut (x <= x + delta * / 2).
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
x <= x + delta * / 2 ->
0 <= f x + - f (x + delta * / 2)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
x <= x + delta * / 2
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 <= / (delta * / 2)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 < l
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
(f x - f (x + delta / 2)) / (delta / 2) + l =
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  intro; generalize (H0 x (x + delta * / 2) H13); intro;
    generalize
      (Rplus_le_compat_l (- f (x + delta / 2)) (f (x + delta / 2)) (f x) H14);
      rewrite Rplus_opp_l; rewrite Rplus_comm; intro; assumption.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
x <= x + delta * / 2
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 <= / (delta * / 2)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 < l
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
(f x - f (x + delta / 2)) / (delta / 2) + l =
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 <= / (delta * / 2)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 < l
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
(f x - f (x + delta / 2)) / (delta / 2) + l =
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  left; apply Rinv_0_lt_compat; assumption.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
0 < l
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
(f x - f (x + delta / 2)) / (delta / 2) + l =
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  assumption.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
(f x - f (x + delta / 2)) / (delta / 2) + l =
- ((f (x + delta / 2) - f x) / (delta / 2) - l)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  rewrite Ropp_minus_distr.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
(f x - f (x + delta / 2)) / (delta / 2) + l =
l - (f (x + delta / 2) - f x) / (delta / 2)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  unfold Rminus.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
(f x + - f (x + delta / 2)) / (delta / 2) + l =
l + - ((f (x + delta / 2) + - f x) / (delta / 2))
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  rewrite (Rplus_comm l).
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
(f x + - f (x + delta / 2)) / (delta / 2) + l =
- ((f (x + delta / 2) + - f x) / (delta / 2)) + l
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  unfold Rdiv.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
(f x + - f (x + delta * / 2)) * / (delta * / 2) + l =
- ((f (x + delta * / 2) + - f x) * / (delta * / 2)) +
l
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  rewrite <- Ropp_mult_distr_l_reverse.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
(f x + - f (x + delta * / 2)) * / (delta * / 2) + l =
- (f (x + delta * / 2) + - f x) * / (delta * / 2) + l
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  rewrite Ropp_plus_distr.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
(f x + - f (x + delta * / 2)) * / (delta * / 2) + l =
(- f (x + delta * / 2) + - - f x) * / (delta * / 2) +
l
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  rewrite Ropp_involutive.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
(f x + - f (x + delta * / 2)) * / (delta * / 2) + l =
(- f (x + delta * / 2) + f x) * / (delta * / 2) + l
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  rewrite (Rplus_comm (f x)).
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:(f (x + delta / 2) - f x) / (delta / 2) <= 0
(- f (x + delta * / 2) + f x) * / (delta * / 2) + l =
(- f (x + delta * / 2) + f x) * / (delta * / 2) + l
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  reflexivity.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta / 2) - f x) / (delta / 2) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  replace ((f (x + delta / 2) - f x) / (delta / 2)) with
  (- ((f x - f (x + delta / 2)) / (delta / 2))).
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
- ((f x - f (x + delta / 2)) / (delta / 2)) <= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
- ((f x - f (x + delta / 2)) / (delta / 2)) =
(f (x + delta / 2) - f x) / (delta / 2)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  rewrite <- Ropp_0.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
- ((f x - f (x + delta / 2)) / (delta / 2)) <= - 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
- ((f x - f (x + delta / 2)) / (delta / 2)) =
(f (x + delta / 2) - f x) / (delta / 2)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  apply Ropp_ge_le_contravar.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f x - f (x + delta / 2)) / (delta / 2) >= 0
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
- ((f x - f (x + delta / 2)) / (delta / 2)) =
(f (x + delta / 2) - f x) / (delta / 2)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  apply Rle_ge.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
0 <= (f x - f (x + delta / 2)) / (delta / 2)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
- ((f x - f (x + delta / 2)) / (delta / 2)) =
(f (x + delta / 2) - f x) / (delta / 2)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  unfold Rdiv; apply Rmult_le_pos.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
0 <= f x - f (x + delta * / 2)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
0 <= / (delta * / 2)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
- ((f x - f (x + delta / 2)) / (delta / 2)) =
(f (x + delta / 2) - f x) / (delta / 2)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  cut (x <= x + delta * / 2).
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
x <= x + delta * / 2 -> 0 <= f x - f (x + delta * / 2)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
x <= x + delta * / 2
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
0 <= / (delta * / 2)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
- ((f x - f (x + delta / 2)) / (delta / 2)) =
(f (x + delta / 2) - f x) / (delta / 2)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  intro; generalize (H0 x (x + delta * / 2) H10); intro.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
H10:x <= x + delta * / 2
H13:f (x + delta * / 2) <= f x
0 <= f x - f (x + delta * / 2)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
x <= x + delta * / 2
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
0 <= / (delta * / 2)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
- ((f x - f (x + delta / 2)) / (delta / 2)) =
(f (x + delta / 2) - f x) / (delta / 2)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  generalize
    (Rplus_le_compat_l (- f (x + delta / 2)) (f (x + delta / 2)) (f x) H13);
    rewrite Rplus_opp_l; rewrite Rplus_comm; intro; assumption.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
x <= x + delta * / 2
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
0 <= / (delta * / 2)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
- ((f x - f (x + delta / 2)) / (delta / 2)) =
(f (x + delta / 2) - f x) / (delta / 2)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
0 <= / (delta * / 2)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
- ((f x - f (x + delta / 2)) / (delta / 2)) =
(f (x + delta / 2) - f x) / (delta / 2)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  left; apply Rinv_0_lt_compat; assumption.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
- ((f x - f (x + delta / 2)) / (delta / 2)) =
(f (x + delta / 2) - f x) / (delta / 2)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  unfold Rdiv; rewrite <- Ropp_mult_distr_l_reverse.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
- (f x - f (x + delta * / 2)) * / (delta * / 2) =
(f (x + delta * / 2) - f x) * / (delta * / 2)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  rewrite Ropp_minus_distr.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
H8:delta / 2 <> 0 /\
0 < delta / 2 /\ Rabs (delta / 2) < delta
H9:delta / 2 <> 0
H11:0 < delta / 2
H12:Rabs (delta / 2) < delta
(f (x + delta * / 2) - f x) * / (delta * / 2) =
(f (x + delta * / 2) - f x) * / (delta * / 2)
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  reflexivity.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  split.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  unfold Rdiv; apply prod_neq_R0.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  generalize (cond_pos delta); intro; red; intro H9; rewrite H9 in H8;
    elim (Rlt_irrefl 0 H8).
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  apply Rinv_neq_0_compat; discrR.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  split.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  rewrite Rabs_right.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  prove_sup0.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
1 * delta < 2 * delta
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  rewrite Rmult_1_l; rewrite double; pattern (pos delta) at 1;
    rewrite <- Rplus_0_r.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  discrR.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  apply Rle_ge; unfold Rdiv; left; apply Rmult_lt_0_compat.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7:forall h : R,
h <> 0 ->
Rabs h < delta ->
Rabs ((f (x + h) - f x) / h - l) < l / 2
0 < delta
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  apply (cond_pos delta).
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
H6:0 < l / 2
delta:posreal
H7: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:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  apply Rinv_0_lt_compat; prove_sup0.
f:R -> R
pr:derivable f
H:decreasing f
x:R
H0:forall x0 y : R, x0 <= y -> f y <= f x0
H1:exists l0 : R, derive_pt f x (pr x) = l0
l:R
H2:derive_pt f x (pr x) = l
H3:l = 0 \/ l > 0
H4:l > 0
H5:derivable_pt_lim f x l
0 < l / 2
  unfold Rdiv; apply Rmult_lt_0_compat;
    [ apply H4 | apply Rinv_0_lt_compat; prove_sup0 ].
Qed.

(**********)
Lemma increasing_decreasing_opp :
  forall f:R -> R, increasing f -> decreasing (- f)%F.
forall f : R -> R, increasing f -> decreasing (- f)%F
Proof.
forall f : R -> R, increasing f -> decreasing (- f)%F
  unfold increasing, decreasing, opp_fct; intros; generalize (H x y H0);
    intro; apply Ropp_ge_le_contravar; apply Rle_ge; assumption.
Qed.

(**********)
Lemma nonpos_derivative_1 :
  forall (f:R -> R) (pr:derivable f),
    (forall x:R, derive_pt f x (pr x) <= 0) -> decreasing f.
forall (f : R -> R) (pr : derivable f),
(forall x : R, derive_pt f x (pr x) <= 0) ->
decreasing f
Proof.
forall (f : R -> R) (pr : derivable f),
(forall x : R, derive_pt f x (pr x) <= 0) ->
decreasing f
  intros.
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) <= 0
decreasing f
  cut (forall h:R, - - f h = f h).
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) <= 0
(forall h : R, - - f h = f h) -> decreasing f
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) <= 0
forall h : R, - - f h = f h
  intro.
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) <= 0
H0:forall h : R, - - f h = f h
decreasing f
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) <= 0
forall h : R, - - f h = f h
  generalize (increasing_decreasing_opp (- f)%F).
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) <= 0
H0:forall h : R, - - f h = f h
(increasing (- f)%F -> decreasing (- - f)%F) ->
decreasing f
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) <= 0
forall h : R, - - f h = f h
  unfold decreasing.
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) <= 0
H0:forall h : R, - - f h = f h
(increasing (- f)%F ->
 forall x y : R, x <= y -> (- - f)%F y <= (- - f)%F x) ->
forall x y : R, x <= y -> f y <= f x
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) <= 0
forall h : R, - - f h = f h
  unfold opp_fct.
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) <= 0
H0:forall h : R, - - f h = f h
(increasing (fun x : R => - f x) ->
 forall x y : R, x <= y -> - - f y <= - - f x) ->
forall x y : R, x <= y -> f y <= f x
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) <= 0
forall h : R, - - f h = f h
  intros.
f:R -> R
pr:derivable f
H:forall x0 : R, derive_pt f x0 (pr x0) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x0 : R => - f x0) ->
forall x0 y0 : R,
x0 <= y0 -> - - f y0 <= - - f x0
x, y:R
H2:x <= y
f y <= f x
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) <= 0
forall h : R, - - f h = f h
  rewrite <- (H0 x); rewrite <- (H0 y).
f:R -> R
pr:derivable f
H:forall x0 : R, derive_pt f x0 (pr x0) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x0 : R => - f x0) ->
forall x0 y0 : R,
x0 <= y0 -> - - f y0 <= - - f x0
x, y:R
H2:x <= y
- - f y <= - - f x
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) <= 0
forall h : R, - - f h = f h
  apply H1.
f:R -> R
pr:derivable f
H:forall x0 : R, derive_pt f x0 (pr x0) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x0 : R => - f x0) ->
forall x0 y0 : R,
x0 <= y0 -> - - f y0 <= - - f x0
x, y:R
H2:x <= y
increasing (fun x0 : R => - f x0)
f:R -> R
pr:derivable f
H:forall x0 : R, derive_pt f x0 (pr x0) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x0 : R => - f x0) ->
forall x0 y0 : R,
x0 <= y0 -> - - f y0 <= - - f x0
x, y:R
H2:x <= y
x <= y
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) <= 0
forall h : R, - - f h = f h
  cut (forall x:R, 0 <= derive_pt (- f) x (derivable_opp f pr x)).
f:R -> R
pr:derivable f
H:forall x0 : R, derive_pt f x0 (pr x0) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x0 : R => - f x0) ->
forall x0 y0 : R,
x0 <= y0 -> - - f y0 <= - - f x0
x, y:R
H2:x <= y
(forall x0 : R,
 0 <= derive_pt (- f) x0 (derivable_opp f pr x0)) ->
increasing (fun x0 : R => - f x0)
f:R -> R
pr:derivable f
H:forall x0 : R, derive_pt f x0 (pr x0) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x0 : R => - f x0) ->
forall x0 y0 : R,
x0 <= y0 -> - - f y0 <= - - f x0
x, y:R
H2:x <= y
forall x0 : R,
0 <= derive_pt (- f) x0 (derivable_opp f pr x0)
f:R -> R
pr:derivable f
H:forall x0 : R, derive_pt f x0 (pr x0) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x0 : R => - f x0) ->
forall x0 y0 : R,
x0 <= y0 -> - - f y0 <= - - f x0
x, y:R
H2:x <= y
x <= y
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) <= 0
forall h : R, - - f h = f h
  intros.
f:R -> R
pr:derivable f
H:forall x0 : R, derive_pt f x0 (pr x0) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x0 : R => - f x0) ->
forall x0 y0 : R,
x0 <= y0 -> - - f y0 <= - - f x0
x, y:R
H2:x <= y
H3:forall x0 : R,
0 <= derive_pt (- f) x0 (derivable_opp f pr x0)
increasing (fun x0 : R => - f x0)
f:R -> R
pr:derivable f
H:forall x0 : R, derive_pt f x0 (pr x0) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x0 : R => - f x0) ->
forall x0 y0 : R,
x0 <= y0 -> - - f y0 <= - - f x0
x, y:R
H2:x <= y
forall x0 : R,
0 <= derive_pt (- f) x0 (derivable_opp f pr x0)
f:R -> R
pr:derivable f
H:forall x0 : R, derive_pt f x0 (pr x0) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x0 : R => - f x0) ->
forall x0 y0 : R,
x0 <= y0 -> - - f y0 <= - - f x0
x, y:R
H2:x <= y
x <= y
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) <= 0
forall h : R, - - f h = f h
  replace (fun x:R => - f x) with (- f)%F; [ idtac | reflexivity ].
f:R -> R
pr:derivable f
H:forall x0 : R, derive_pt f x0 (pr x0) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x0 : R => - f x0) ->
forall x0 y0 : R,
x0 <= y0 -> - - f y0 <= - - f x0
x, y:R
H2:x <= y
H3:forall x0 : R,
0 <= derive_pt (- f) x0 (derivable_opp f pr x0)
increasing (- f)%F
f:R -> R
pr:derivable f
H:forall x0 : R, derive_pt f x0 (pr x0) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x0 : R => - f x0) ->
forall x0 y0 : R,
x0 <= y0 -> - - f y0 <= - - f x0
x, y:R
H2:x <= y
forall x0 : R,
0 <= derive_pt (- f) x0 (derivable_opp f pr x0)
f:R -> R
pr:derivable f
H:forall x0 : R, derive_pt f x0 (pr x0) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x0 : R => - f x0) ->
forall x0 y0 : R,
x0 <= y0 -> - - f y0 <= - - f x0
x, y:R
H2:x <= y
x <= y
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) <= 0
forall h : R, - - f h = f h
  apply (nonneg_derivative_1 (- f)%F (derivable_opp f pr) H3).
f:R -> R
pr:derivable f
H:forall x0 : R, derive_pt f x0 (pr x0) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x0 : R => - f x0) ->
forall x0 y0 : R,
x0 <= y0 -> - - f y0 <= - - f x0
x, y:R
H2:x <= y
forall x0 : R,
0 <= derive_pt (- f) x0 (derivable_opp f pr x0)
f:R -> R
pr:derivable f
H:forall x0 : R, derive_pt f x0 (pr x0) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x0 : R => - f x0) ->
forall x0 y0 : R,
x0 <= y0 -> - - f y0 <= - - f x0
x, y:R
H2:x <= y
x <= y
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) <= 0
forall h : R, - - f h = f h
  intro.
f:R -> R
pr:derivable f
H:forall x1 : R, derive_pt f x1 (pr x1) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x1 : R => - f x1) ->
forall x1 y0 : R,
x1 <= y0 -> - - f y0 <= - - f x1
x, y:R
H2:x <= y
x0:R
0 <= derive_pt (- f) x0 (derivable_opp f pr x0)
f:R -> R
pr:derivable f
H:forall x0 : R, derive_pt f x0 (pr x0) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x0 : R => - f x0) ->
forall x0 y0 : R,
x0 <= y0 -> - - f y0 <= - - f x0
x, y:R
H2:x <= y
x <= y
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) <= 0
forall h : R, - - f h = f h
  assert (H3 := derive_pt_opp f x0 (pr x0)).
f:R -> R
pr:derivable f
H:forall x1 : R, derive_pt f x1 (pr x1) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x1 : R => - f x1) ->
forall x1 y0 : R,
x1 <= y0 -> - - f y0 <= - - f x1
x, y:R
H2:x <= y
x0:R
H3:derive_pt (- f) x0
  (derivable_pt_opp f x0 (pr x0)) =
- derive_pt f x0 (pr x0)
0 <= derive_pt (- f) x0 (derivable_opp f pr x0)
f:R -> R
pr:derivable f
H:forall x0 : R, derive_pt f x0 (pr x0) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x0 : R => - f x0) ->
forall x0 y0 : R,
x0 <= y0 -> - - f y0 <= - - f x0
x, y:R
H2:x <= y
x <= y
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) <= 0
forall h : R, - - f h = f h
  cut
    (derive_pt (- f) x0 (derivable_pt_opp f x0 (pr x0)) =
      derive_pt (- f) x0 (derivable_opp f pr x0)).
f:R -> R
pr:derivable f
H:forall x1 : R, derive_pt f x1 (pr x1) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x1 : R => - f x1) ->
forall x1 y0 : R,
x1 <= y0 -> - - f y0 <= - - f x1
x, y:R
H2:x <= y
x0:R
H3:derive_pt (- f) x0
  (derivable_pt_opp f x0 (pr x0)) =
- derive_pt f x0 (pr x0)
derive_pt (- f) x0 (derivable_pt_opp f x0 (pr x0)) =
derive_pt (- f) x0 (derivable_opp f pr x0) ->
0 <= derive_pt (- f) x0 (derivable_opp f pr x0)
f:R -> R
pr:derivable f
H:forall x1 : R, derive_pt f x1 (pr x1) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x1 : R => - f x1) ->
forall x1 y0 : R,
x1 <= y0 -> - - f y0 <= - - f x1
x, y:R
H2:x <= y
x0:R
H3:derive_pt (- f) x0
  (derivable_pt_opp f x0 (pr x0)) =
- derive_pt f x0 (pr x0)
derive_pt (- f) x0 (derivable_pt_opp f x0 (pr x0)) =
derive_pt (- f) x0 (derivable_opp f pr x0)
f:R -> R
pr:derivable f
H:forall x0 : R, derive_pt f x0 (pr x0) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x0 : R => - f x0) ->
forall x0 y0 : R,
x0 <= y0 -> - - f y0 <= - - f x0
x, y:R
H2:x <= y
x <= y
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) <= 0
forall h : R, - - f h = f h
  intro.
f:R -> R
pr:derivable f
H:forall x1 : R, derive_pt f x1 (pr x1) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x1 : R => - f x1) ->
forall x1 y0 : R,
x1 <= y0 -> - - f y0 <= - - f x1
x, y:R
H2:x <= y
x0:R
H3:derive_pt (- f) x0
  (derivable_pt_opp f x0 (pr x0)) =
- derive_pt f x0 (pr x0)
H4:derive_pt (- f) x0
  (derivable_pt_opp f x0 (pr x0)) =
derive_pt (- f) x0 (derivable_opp f pr x0)
0 <= derive_pt (- f) x0 (derivable_opp f pr x0)
f:R -> R
pr:derivable f
H:forall x1 : R, derive_pt f x1 (pr x1) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x1 : R => - f x1) ->
forall x1 y0 : R,
x1 <= y0 -> - - f y0 <= - - f x1
x, y:R
H2:x <= y
x0:R
H3:derive_pt (- f) x0
  (derivable_pt_opp f x0 (pr x0)) =
- derive_pt f x0 (pr x0)
derive_pt (- f) x0 (derivable_pt_opp f x0 (pr x0)) =
derive_pt (- f) x0 (derivable_opp f pr x0)
f:R -> R
pr:derivable f
H:forall x0 : R, derive_pt f x0 (pr x0) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x0 : R => - f x0) ->
forall x0 y0 : R,
x0 <= y0 -> - - f y0 <= - - f x0
x, y:R
H2:x <= y
x <= y
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) <= 0
forall h : R, - - f h = f h
  rewrite <- H4.
f:R -> R
pr:derivable f
H:forall x1 : R, derive_pt f x1 (pr x1) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x1 : R => - f x1) ->
forall x1 y0 : R,
x1 <= y0 -> - - f y0 <= - - f x1
x, y:R
H2:x <= y
x0:R
H3:derive_pt (- f) x0
  (derivable_pt_opp f x0 (pr x0)) =
- derive_pt f x0 (pr x0)
H4:derive_pt (- f) x0
  (derivable_pt_opp f x0 (pr x0)) =
derive_pt (- f) x0 (derivable_opp f pr x0)
0 <=
derive_pt (- f) x0 (derivable_pt_opp f x0 (pr x0))
f:R -> R
pr:derivable f
H:forall x1 : R, derive_pt f x1 (pr x1) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x1 : R => - f x1) ->
forall x1 y0 : R,
x1 <= y0 -> - - f y0 <= - - f x1
x, y:R
H2:x <= y
x0:R
H3:derive_pt (- f) x0
  (derivable_pt_opp f x0 (pr x0)) =
- derive_pt f x0 (pr x0)
derive_pt (- f) x0 (derivable_pt_opp f x0 (pr x0)) =
derive_pt (- f) x0 (derivable_opp f pr x0)
f:R -> R
pr:derivable f
H:forall x0 : R, derive_pt f x0 (pr x0) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x0 : R => - f x0) ->
forall x0 y0 : R,
x0 <= y0 -> - - f y0 <= - - f x0
x, y:R
H2:x <= y
x <= y
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) <= 0
forall h : R, - - f h = f h
  rewrite H3.
f:R -> R
pr:derivable f
H:forall x1 : R, derive_pt f x1 (pr x1) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x1 : R => - f x1) ->
forall x1 y0 : R,
x1 <= y0 -> - - f y0 <= - - f x1
x, y:R
H2:x <= y
x0:R
H3:derive_pt (- f) x0
  (derivable_pt_opp f x0 (pr x0)) =
- derive_pt f x0 (pr x0)
H4:derive_pt (- f) x0
  (derivable_pt_opp f x0 (pr x0)) =
derive_pt (- f) x0 (derivable_opp f pr x0)
0 <= - derive_pt f x0 (pr x0)
f:R -> R
pr:derivable f
H:forall x1 : R, derive_pt f x1 (pr x1) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x1 : R => - f x1) ->
forall x1 y0 : R,
x1 <= y0 -> - - f y0 <= - - f x1
x, y:R
H2:x <= y
x0:R
H3:derive_pt (- f) x0
  (derivable_pt_opp f x0 (pr x0)) =
- derive_pt f x0 (pr x0)
derive_pt (- f) x0 (derivable_pt_opp f x0 (pr x0)) =
derive_pt (- f) x0 (derivable_opp f pr x0)
f:R -> R
pr:derivable f
H:forall x0 : R, derive_pt f x0 (pr x0) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x0 : R => - f x0) ->
forall x0 y0 : R,
x0 <= y0 -> - - f y0 <= - - f x0
x, y:R
H2:x <= y
x <= y
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) <= 0
forall h : R, - - f h = f h
  rewrite <- Ropp_0; apply Ropp_ge_le_contravar; apply Rle_ge; apply (H x0).
f:R -> R
pr:derivable f
H:forall x1 : R, derive_pt f x1 (pr x1) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x1 : R => - f x1) ->
forall x1 y0 : R,
x1 <= y0 -> - - f y0 <= - - f x1
x, y:R
H2:x <= y
x0:R
H3:derive_pt (- f) x0
  (derivable_pt_opp f x0 (pr x0)) =
- derive_pt f x0 (pr x0)
derive_pt (- f) x0 (derivable_pt_opp f x0 (pr x0)) =
derive_pt (- f) x0 (derivable_opp f pr x0)
f:R -> R
pr:derivable f
H:forall x0 : R, derive_pt f x0 (pr x0) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x0 : R => - f x0) ->
forall x0 y0 : R,
x0 <= y0 -> - - f y0 <= - - f x0
x, y:R
H2:x <= y
x <= y
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) <= 0
forall h : R, - - f h = f h
  apply pr_nu.
f:R -> R
pr:derivable f
H:forall x0 : R, derive_pt f x0 (pr x0) <= 0
H0:forall h : R, - - f h = f h
H1:increasing (fun x0 : R => - f x0) ->
forall x0 y0 : R,
x0 <= y0 -> - - f y0 <= - - f x0
x, y:R
H2:x <= y
x <= y
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) <= 0
forall h : R, - - f h = f h
  assumption.
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) <= 0
forall h : R, - - f h = f h
  intro; ring.
Qed.

(**********)
Lemma positive_derivative :
  forall (f:R -> R) (pr:derivable f),
    (forall x:R, 0 < derive_pt f x (pr x)) -> strict_increasing f.
forall (f : R -> R) (pr : derivable f),
(forall x : R, 0 < derive_pt f x (pr x)) ->
strict_increasing f
Proof.
forall (f : R -> R) (pr : derivable f),
(forall x : R, 0 < derive_pt f x (pr x)) ->
strict_increasing f
  intros.
f:R -> R
pr:derivable f
H:forall x : R, 0 < derive_pt f x (pr x)
strict_increasing f
  unfold strict_increasing.
f:R -> R
pr:derivable f
H:forall x : R, 0 < derive_pt f x (pr x)
forall x y : R, x < y -> f x < f y
  intros.
f:R -> R
pr:derivable f
H:forall x0 : R, 0 < derive_pt f x0 (pr x0)
x, y:R
H0:x < y
f x < f y
  apply Rplus_lt_reg_l with (- f x).
f:R -> R
pr:derivable f
H:forall x0 : R, 0 < derive_pt f x0 (pr x0)
x, y:R
H0:x < y
- f x + f x < - f x + f y
  rewrite Rplus_opp_l; rewrite Rplus_comm.
f:R -> R
pr:derivable f
H:forall x0 : R, 0 < derive_pt f x0 (pr x0)
x, y:R
H0:x < y
0 < f y + - f x
  assert (H1 := MVT_cor1 f _ _ pr H0).
f:R -> R
pr:derivable f
H:forall x0 : R, 0 < derive_pt f x0 (pr x0)
x, y:R
H0:x < y
H1:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
0 < f y + - f x
  elim H1; intros.
f:R -> R
pr:derivable f
H:forall x1 : R, 0 < derive_pt f x1 (pr x1)
x, y:R
H0:x < y
H1:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H2:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
0 < f y + - f x
  elim H2; intros.
f:R -> R
pr:derivable f
H:forall x1 : R, 0 < derive_pt f x1 (pr x1)
x, y:R
H0:x < y
H1:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H2:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H3:f y - f x = derive_pt f x0 (pr x0) * (y - x)
H4:x < x0 < y
0 < f y + - f x
  unfold Rminus in H3.
f:R -> R
pr:derivable f
H:forall x1 : R, 0 < derive_pt f x1 (pr x1)
x, y:R
H0:x < y
H1:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H2:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H3:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H4:x < x0 < y
0 < f y + - f x
  rewrite H3.
f:R -> R
pr:derivable f
H:forall x1 : R, 0 < derive_pt f x1 (pr x1)
x, y:R
H0:x < y
H1:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H2:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H3:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H4:x < x0 < y
0 < derive_pt f x0 (pr x0) * (y + - x)
  apply Rmult_lt_0_compat.
f:R -> R
pr:derivable f
H:forall x1 : R, 0 < derive_pt f x1 (pr x1)
x, y:R
H0:x < y
H1:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H2:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H3:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H4:x < x0 < y
0 < derive_pt f x0 (pr x0)
f:R -> R
pr:derivable f
H:forall x1 : R, 0 < derive_pt f x1 (pr x1)
x, y:R
H0:x < y
H1:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H2:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H3:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H4:x < x0 < y
0 < y + - x
  apply H.
f:R -> R
pr:derivable f
H:forall x1 : R, 0 < derive_pt f x1 (pr x1)
x, y:R
H0:x < y
H1:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H2:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H3:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H4:x < x0 < y
0 < y + - x
  apply Rplus_lt_reg_l with x.
f:R -> R
pr:derivable f
H:forall x1 : R, 0 < derive_pt f x1 (pr x1)
x, y:R
H0:x < y
H1:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H2:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H3:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H4:x < x0 < y
x + 0 < x + (y + - x)
  rewrite Rplus_0_r; replace (x + (y + - x)) with y; [ assumption | ring ].
Qed.

(**********)
Lemma strictincreasing_strictdecreasing_opp :
  forall f:R -> R, strict_increasing f -> strict_decreasing (- f)%F.
forall f : R -> R,
strict_increasing f -> strict_decreasing (- f)%F
Proof.
forall f : R -> R,
strict_increasing f -> strict_decreasing (- f)%F
  unfold strict_increasing, strict_decreasing, opp_fct; intros;
    generalize (H x y H0); intro; apply Ropp_lt_gt_contravar;
      assumption.
Qed.

(**********)
Lemma negative_derivative :
  forall (f:R -> R) (pr:derivable f),
    (forall x:R, derive_pt f x (pr x) < 0) -> strict_decreasing f.
forall (f : R -> R) (pr : derivable f),
(forall x : R, derive_pt f x (pr x) < 0) ->
strict_decreasing f
Proof.
forall (f : R -> R) (pr : derivable f),
(forall x : R, derive_pt f x (pr x) < 0) ->
strict_decreasing f
  intros.
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) < 0
strict_decreasing f
  cut (forall h:R, - - f h = f h).
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) < 0
(forall h : R, - - f h = f h) -> strict_decreasing f
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) < 0
forall h : R, - - f h = f h
  intros.
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) < 0
H0:forall h : R, - - f h = f h
strict_decreasing f
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) < 0
forall h : R, - - f h = f h
  generalize (strictincreasing_strictdecreasing_opp (- f)%F).
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) < 0
H0:forall h : R, - - f h = f h
(strict_increasing (- f)%F ->
 strict_decreasing (- - f)%F) -> strict_decreasing f
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) < 0
forall h : R, - - f h = f h
  unfold strict_decreasing, opp_fct.
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) < 0
H0:forall h : R, - - f h = f h
(strict_increasing (fun x : R => - f x) ->
 forall x y : R, x < y -> - - f y < - - f x) ->
forall x y : R, x < y -> f y < f x
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) < 0
forall h : R, - - f h = f h
  intros.
f:R -> R
pr:derivable f
H:forall x0 : R, derive_pt f x0 (pr x0) < 0
H0:forall h : R, - - f h = f h
H1:strict_increasing (fun x0 : R => - f x0) ->
forall x0 y0 : R, x0 < y0 -> - - f y0 < - - f x0
x, y:R
H2:x < y
f y < f x
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) < 0
forall h : R, - - f h = f h
  rewrite <- (H0 x).
f:R -> R
pr:derivable f
H:forall x0 : R, derive_pt f x0 (pr x0) < 0
H0:forall h : R, - - f h = f h
H1:strict_increasing (fun x0 : R => - f x0) ->
forall x0 y0 : R, x0 < y0 -> - - f y0 < - - f x0
x, y:R
H2:x < y
f y < - - f x
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) < 0
forall h : R, - - f h = f h
  rewrite <- (H0 y).
f:R -> R
pr:derivable f
H:forall x0 : R, derive_pt f x0 (pr x0) < 0
H0:forall h : R, - - f h = f h
H1:strict_increasing (fun x0 : R => - f x0) ->
forall x0 y0 : R, x0 < y0 -> - - f y0 < - - f x0
x, y:R
H2:x < y
- - f y < - - f x
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) < 0
forall h : R, - - f h = f h
  apply H1; [ idtac | assumption ].
f:R -> R
pr:derivable f
H:forall x0 : R, derive_pt f x0 (pr x0) < 0
H0:forall h : R, - - f h = f h
H1:strict_increasing (fun x0 : R => - f x0) ->
forall x0 y0 : R, x0 < y0 -> - - f y0 < - - f x0
x, y:R
H2:x < y
strict_increasing (fun x0 : R => - f x0)
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) < 0
forall h : R, - - f h = f h
  cut (forall x:R, 0 < derive_pt (- f) x (derivable_opp f pr x)).
f:R -> R
pr:derivable f
H:forall x0 : R, derive_pt f x0 (pr x0) < 0
H0:forall h : R, - - f h = f h
H1:strict_increasing (fun x0 : R => - f x0) ->
forall x0 y0 : R, x0 < y0 -> - - f y0 < - - f x0
x, y:R
H2:x < y
(forall x0 : R,
 0 < derive_pt (- f) x0 (derivable_opp f pr x0)) ->
strict_increasing (fun x0 : R => - f x0)
f:R -> R
pr:derivable f
H:forall x0 : R, derive_pt f x0 (pr x0) < 0
H0:forall h : R, - - f h = f h
H1:strict_increasing (fun x0 : R => - f x0) ->
forall x0 y0 : R, x0 < y0 -> - - f y0 < - - f x0
x, y:R
H2:x < y
forall x0 : R,
0 < derive_pt (- f) x0 (derivable_opp f pr x0)
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) < 0
forall h : R, - - f h = f h
  intros; eapply positive_derivative; apply H3.
f:R -> R
pr:derivable f
H:forall x0 : R, derive_pt f x0 (pr x0) < 0
H0:forall h : R, - - f h = f h
H1:strict_increasing (fun x0 : R => - f x0) ->
forall x0 y0 : R, x0 < y0 -> - - f y0 < - - f x0
x, y:R
H2:x < y
forall x0 : R,
0 < derive_pt (- f) x0 (derivable_opp f pr x0)
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) < 0
forall h : R, - - f h = f h
  intro.
f:R -> R
pr:derivable f
H:forall x1 : R, derive_pt f x1 (pr x1) < 0
H0:forall h : R, - - f h = f h
H1:strict_increasing (fun x1 : R => - f x1) ->
forall x1 y0 : R, x1 < y0 -> - - f y0 < - - f x1
x, y:R
H2:x < y
x0:R
0 < derive_pt (- f) x0 (derivable_opp f pr x0)
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) < 0
forall h : R, - - f h = f h
  assert (H3 := derive_pt_opp f x0 (pr x0)).
f:R -> R
pr:derivable f
H:forall x1 : R, derive_pt f x1 (pr x1) < 0
H0:forall h : R, - - f h = f h
H1:strict_increasing (fun x1 : R => - f x1) ->
forall x1 y0 : R, x1 < y0 -> - - f y0 < - - f x1
x, y:R
H2:x < y
x0:R
H3:derive_pt (- f) x0
  (derivable_pt_opp f x0 (pr x0)) =
- derive_pt f x0 (pr x0)
0 < derive_pt (- f) x0 (derivable_opp f pr x0)
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) < 0
forall h : R, - - f h = f h
  cut
    (derive_pt (- f) x0 (derivable_pt_opp f x0 (pr x0)) =
      derive_pt (- f) x0 (derivable_opp f pr x0)).
f:R -> R
pr:derivable f
H:forall x1 : R, derive_pt f x1 (pr x1) < 0
H0:forall h : R, - - f h = f h
H1:strict_increasing (fun x1 : R => - f x1) ->
forall x1 y0 : R, x1 < y0 -> - - f y0 < - - f x1
x, y:R
H2:x < y
x0:R
H3:derive_pt (- f) x0
  (derivable_pt_opp f x0 (pr x0)) =
- derive_pt f x0 (pr x0)
derive_pt (- f) x0 (derivable_pt_opp f x0 (pr x0)) =
derive_pt (- f) x0 (derivable_opp f pr x0) ->
0 < derive_pt (- f) x0 (derivable_opp f pr x0)
f:R -> R
pr:derivable f
H:forall x1 : R, derive_pt f x1 (pr x1) < 0
H0:forall h : R, - - f h = f h
H1:strict_increasing (fun x1 : R => - f x1) ->
forall x1 y0 : R, x1 < y0 -> - - f y0 < - - f x1
x, y:R
H2:x < y
x0:R
H3:derive_pt (- f) x0
  (derivable_pt_opp f x0 (pr x0)) =
- derive_pt f x0 (pr x0)
derive_pt (- f) x0 (derivable_pt_opp f x0 (pr x0)) =
derive_pt (- f) x0 (derivable_opp f pr x0)
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) < 0
forall h : R, - - f h = f h
  intro.
f:R -> R
pr:derivable f
H:forall x1 : R, derive_pt f x1 (pr x1) < 0
H0:forall h : R, - - f h = f h
H1:strict_increasing (fun x1 : R => - f x1) ->
forall x1 y0 : R, x1 < y0 -> - - f y0 < - - f x1
x, y:R
H2:x < y
x0:R
H3:derive_pt (- f) x0
  (derivable_pt_opp f x0 (pr x0)) =
- derive_pt f x0 (pr x0)
H4:derive_pt (- f) x0
  (derivable_pt_opp f x0 (pr x0)) =
derive_pt (- f) x0 (derivable_opp f pr x0)
0 < derive_pt (- f) x0 (derivable_opp f pr x0)
f:R -> R
pr:derivable f
H:forall x1 : R, derive_pt f x1 (pr x1) < 0
H0:forall h : R, - - f h = f h
H1:strict_increasing (fun x1 : R => - f x1) ->
forall x1 y0 : R, x1 < y0 -> - - f y0 < - - f x1
x, y:R
H2:x < y
x0:R
H3:derive_pt (- f) x0
  (derivable_pt_opp f x0 (pr x0)) =
- derive_pt f x0 (pr x0)
derive_pt (- f) x0 (derivable_pt_opp f x0 (pr x0)) =
derive_pt (- f) x0 (derivable_opp f pr x0)
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) < 0
forall h : R, - - f h = f h
  rewrite <- H4; rewrite H3.
f:R -> R
pr:derivable f
H:forall x1 : R, derive_pt f x1 (pr x1) < 0
H0:forall h : R, - - f h = f h
H1:strict_increasing (fun x1 : R => - f x1) ->
forall x1 y0 : R, x1 < y0 -> - - f y0 < - - f x1
x, y:R
H2:x < y
x0:R
H3:derive_pt (- f) x0
  (derivable_pt_opp f x0 (pr x0)) =
- derive_pt f x0 (pr x0)
H4:derive_pt (- f) x0
  (derivable_pt_opp f x0 (pr x0)) =
derive_pt (- f) x0 (derivable_opp f pr x0)
0 < - derive_pt f x0 (pr x0)
f:R -> R
pr:derivable f
H:forall x1 : R, derive_pt f x1 (pr x1) < 0
H0:forall h : R, - - f h = f h
H1:strict_increasing (fun x1 : R => - f x1) ->
forall x1 y0 : R, x1 < y0 -> - - f y0 < - - f x1
x, y:R
H2:x < y
x0:R
H3:derive_pt (- f) x0
  (derivable_pt_opp f x0 (pr x0)) =
- derive_pt f x0 (pr x0)
derive_pt (- f) x0 (derivable_pt_opp f x0 (pr x0)) =
derive_pt (- f) x0 (derivable_opp f pr x0)
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) < 0
forall h : R, - - f h = f h
  rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; apply (H x0).
f:R -> R
pr:derivable f
H:forall x1 : R, derive_pt f x1 (pr x1) < 0
H0:forall h : R, - - f h = f h
H1:strict_increasing (fun x1 : R => - f x1) ->
forall x1 y0 : R, x1 < y0 -> - - f y0 < - - f x1
x, y:R
H2:x < y
x0:R
H3:derive_pt (- f) x0
  (derivable_pt_opp f x0 (pr x0)) =
- derive_pt f x0 (pr x0)
derive_pt (- f) x0 (derivable_pt_opp f x0 (pr x0)) =
derive_pt (- f) x0 (derivable_opp f pr x0)
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) < 0
forall h : R, - - f h = f h
  apply pr_nu.
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) < 0
forall h : R, - - f h = f h
  intro; ring.
Qed.

(**********)
Lemma null_derivative_0 :
  forall (f:R -> R) (pr:derivable f),
    constant f -> forall x:R, derive_pt f x (pr x) = 0.
forall (f : R -> R) (pr : derivable f),
constant f -> forall x : R, derive_pt f x (pr x) = 0
Proof.
forall (f : R -> R) (pr : derivable f),
constant f -> forall x : R, derive_pt f x (pr x) = 0
  intros.
f:R -> R
pr:derivable f
H:constant f
x:R
derive_pt f x (pr x) = 0
  unfold constant in H.
f:R -> R
pr:derivable f
H:forall x0 y : R, f x0 = f y
x:R
derive_pt f x (pr x) = 0
  apply derive_pt_eq_0.
f:R -> R
pr:derivable f
H:forall x0 y : R, f x0 = f y
x:R
derivable_pt_lim f x 0
  intros; exists (mkposreal 1 Rlt_0_1); simpl; intros.
f:R -> R
pr:derivable f
H:forall x0 y : R, f x0 = f y
x, eps:R
H0:0 < eps
h:R
H1:h <> 0
H2:Rabs h < 1
Rabs ((f (x + h) - f x) / h - 0) < eps
  rewrite (H x (x + h)); unfold Rminus; unfold Rdiv;
    rewrite Rplus_opp_r; rewrite Rmult_0_l; rewrite Rplus_opp_r;
      rewrite Rabs_R0; assumption.
Qed.

(**********)
Lemma increasing_decreasing :
  forall f:R -> R, increasing f -> decreasing f -> constant f.
forall f : R -> R,
increasing f -> decreasing f -> constant f
Proof.
forall f : R -> R,
increasing f -> decreasing f -> constant f
  unfold increasing, decreasing, constant; intros;
    case (Rtotal_order x y); intro.
f:R -> R
H:forall x0 y0 : R, x0 <= y0 -> f x0 <= f y0
H0:forall x0 y0 : R, x0 <= y0 -> f y0 <= f x0
x, y:R
H1:x < y
f x = f y
f:R -> R
H:forall x0 y0 : R, x0 <= y0 -> f x0 <= f y0
H0:forall x0 y0 : R, x0 <= y0 -> f y0 <= f x0
x, y:R
H1:x = y \/ x > y
f x = f y
  generalize (Rlt_le x y H1); intro;
    apply (Rle_antisym (f x) (f y) (H x y H2) (H0 x y H2)).
f:R -> R
H:forall x0 y0 : R, x0 <= y0 -> f x0 <= f y0
H0:forall x0 y0 : R, x0 <= y0 -> f y0 <= f x0
x, y:R
H1:x = y \/ x > y
f x = f y
  elim H1; intro.
f:R -> R
H:forall x0 y0 : R, x0 <= y0 -> f x0 <= f y0
H0:forall x0 y0 : R, x0 <= y0 -> f y0 <= f x0
x, y:R
H1:x = y \/ x > y
H2:x = y
f x = f y
f:R -> R
H:forall x0 y0 : R, x0 <= y0 -> f x0 <= f y0
H0:forall x0 y0 : R, x0 <= y0 -> f y0 <= f x0
x, y:R
H1:x = y \/ x > y
H2:x > y
f x = f y
  rewrite H2; reflexivity.
f:R -> R
H:forall x0 y0 : R, x0 <= y0 -> f x0 <= f y0
H0:forall x0 y0 : R, x0 <= y0 -> f y0 <= f x0
x, y:R
H1:x = y \/ x > y
H2:x > y
f x = f y
  generalize (Rlt_le y x H2); intro; symmetry ;
    apply (Rle_antisym (f y) (f x) (H y x H3) (H0 y x H3)).
Qed.

(**********)
Lemma null_derivative_1 :
  forall (f:R -> R) (pr:derivable f),
    (forall x:R, derive_pt f x (pr x) = 0) -> constant f.
forall (f : R -> R) (pr : derivable f),
(forall x : R, derive_pt f x (pr x) = 0) -> constant f
Proof.
forall (f : R -> R) (pr : derivable f),
(forall x : R, derive_pt f x (pr x) = 0) -> constant f
  intros.
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) = 0
constant f
  cut (forall x:R, derive_pt f x (pr x) <= 0).
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) = 0
(forall x : R, derive_pt f x (pr x) <= 0) ->
constant f
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) = 0
forall x : R, derive_pt f x (pr x) <= 0
  cut (forall x:R, 0 <= derive_pt f x (pr x)).
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) = 0
(forall x : R, 0 <= derive_pt f x (pr x)) ->
(forall x : R, derive_pt f x (pr x) <= 0) ->
constant f
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) = 0
forall x : R, 0 <= derive_pt f x (pr x)
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) = 0
forall x : R, derive_pt f x (pr x) <= 0
  intros.
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) = 0
H0:forall x : R, 0 <= derive_pt f x (pr x)
H1:forall x : R, derive_pt f x (pr x) <= 0
constant f
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) = 0
forall x : R, 0 <= derive_pt f x (pr x)
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) = 0
forall x : R, derive_pt f x (pr x) <= 0
  assert (H2 := nonneg_derivative_1 f pr H0).
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) = 0
H0:forall x : R, 0 <= derive_pt f x (pr x)
H1:forall x : R, derive_pt f x (pr x) <= 0
H2:increasing f
constant f
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) = 0
forall x : R, 0 <= derive_pt f x (pr x)
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) = 0
forall x : R, derive_pt f x (pr x) <= 0
  assert (H3 := nonpos_derivative_1 f pr H1).
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) = 0
H0:forall x : R, 0 <= derive_pt f x (pr x)
H1:forall x : R, derive_pt f x (pr x) <= 0
H2:increasing f
H3:decreasing f
constant f
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) = 0
forall x : R, 0 <= derive_pt f x (pr x)
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) = 0
forall x : R, derive_pt f x (pr x) <= 0
  apply increasing_decreasing; assumption.
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) = 0
forall x : R, 0 <= derive_pt f x (pr x)
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) = 0
forall x : R, derive_pt f x (pr x) <= 0
  intro; right; symmetry ; apply (H x).
f:R -> R
pr:derivable f
H:forall x : R, derive_pt f x (pr x) = 0
forall x : R, derive_pt f x (pr x) <= 0
  intro; right; apply (H x).
Qed.

(**********)
Lemma derive_increasing_interv_ax :
  forall (a b:R) (f:R -> R) (pr:derivable f),
    a < b ->
    ((forall t:R, a < t < b -> 0 < derive_pt f t (pr t)) ->
      forall x y:R, a <= x <= b -> a <= y <= b -> x < y -> f x < f y) /\
    ((forall t:R, a < t < b -> 0 <= derive_pt f t (pr t)) ->
      forall x y:R, a <= x <= b -> a <= y <= b -> x < y -> f x <= f y).
forall (a b : R) (f : R -> R) (pr : derivable f),
a < b ->
((forall t : R, a < t < b -> 0 < derive_pt f t (pr t)) ->
 forall x y : R,
 a <= x <= b -> a <= y <= b -> x < y -> f x < f y) /\
((forall t : R, a < t < b -> 0 <= derive_pt f t (pr t)) ->
 forall x y : R,
 a <= x <= b -> a <= y <= b -> x < y -> f x <= f y)
Proof.
forall (a b : R) (f : R -> R) (pr : derivable f),
a < b ->
((forall t : R, a < t < b -> 0 < derive_pt f t (pr t)) ->
 forall x y : R,
 a <= x <= b -> a <= y <= b -> x < y -> f x < f y) /\
((forall t : R, a < t < b -> 0 <= derive_pt f t (pr t)) ->
 forall x y : R,
 a <= x <= b -> a <= y <= b -> x < y -> f x <= f y)
  intros.
a, b:R
f:R -> R
pr:derivable f
H:a < b
((forall t : R, a < t < b -> 0 < derive_pt f t (pr t)) ->
 forall x y : R,
 a <= x <= b -> a <= y <= b -> x < y -> f x < f y) /\
((forall t : R, a < t < b -> 0 <= derive_pt f t (pr t)) ->
 forall x y : R,
 a <= x <= b -> a <= y <= b -> x < y -> f x <= f y)
  split; intros.
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 < derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
f x < f y
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
f x <= f y
  apply Rplus_lt_reg_l with (- f x).
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 < derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
- f x + f x < - f x + f y
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
f x <= f y
  rewrite Rplus_opp_l; rewrite Rplus_comm.
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 < derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
0 < f y + - f x
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
f x <= f y
  assert (H4 := MVT_cor1 f _ _ pr H3).
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 < derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
0 < f y + - f x
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
f x <= f y
  elim H4; intros.
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 < derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
0 < f y + - f x
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
f x <= f y
  elim H5; intros.
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 < derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y - f x = derive_pt f x0 (pr x0) * (y - x)
H7:x < x0 < y
0 < f y + - f x
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
f x <= f y
  unfold Rminus in H6.
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 < derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
0 < f y + - f x
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
f x <= f y
  rewrite H6.
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 < derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
0 < derive_pt f x0 (pr x0) * (y + - x)
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
f x <= f y
  apply Rmult_lt_0_compat.
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 < derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
0 < derive_pt f x0 (pr x0)
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 < derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
0 < y + - x
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
f x <= f y
  apply H0.
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 < derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
a < x0 < b
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 < derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
0 < y + - x
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
f x <= f y
  elim H7; intros.
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 < derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
H8:x < x0
H9:x0 < y
a < x0 < b
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 < derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
0 < y + - x
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
f x <= f y
  split.
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 < derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
H8:x < x0
H9:x0 < y
a < x0
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 < derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
H8:x < x0
H9:x0 < y
x0 < b
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 < derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
0 < y + - x
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
f x <= f y
  elim H1; intros.
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 < derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
H8:x < x0
H9:x0 < y
H10:a <= x
H11:x <= b
a < x0
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 < derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
H8:x < x0
H9:x0 < y
x0 < b
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 < derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
0 < y + - x
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
f x <= f y
  apply Rle_lt_trans with x; assumption.
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 < derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
H8:x < x0
H9:x0 < y
x0 < b
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 < derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
0 < y + - x
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
f x <= f y
  elim H2; intros.
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 < derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
H8:x < x0
H9:x0 < y
H10:a <= y
H11:y <= b
x0 < b
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 < derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
0 < y + - x
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
f x <= f y
  apply Rlt_le_trans with y; assumption.
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 < derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
0 < y + - x
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
f x <= f y
  apply Rplus_lt_reg_l with x.
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 < derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
x + 0 < x + (y + - x)
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
f x <= f y
  rewrite Rplus_0_r; replace (x + (y + - x)) with y; [ assumption | ring ].
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
f x <= f y
  apply Rplus_le_reg_l with (- f x).
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
- f x + f x <= - f x + f y
  rewrite Rplus_opp_l; rewrite Rplus_comm.
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
0 <= f y + - f x
  assert (H4 := MVT_cor1 f _ _ pr H3).
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
0 <= f y + - f x
  elim H4; intros.
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
0 <= f y + - f x
  elim H5; intros.
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y - f x = derive_pt f x0 (pr x0) * (y - x)
H7:x < x0 < y
0 <= f y + - f x
  unfold Rminus in H6.
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
0 <= f y + - f x
  rewrite H6.
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
0 <= derive_pt f x0 (pr x0) * (y + - x)
  apply Rmult_le_pos.
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
0 <= derive_pt f x0 (pr x0)
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
0 <= y + - x
  apply H0.
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
a < x0 < b
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
0 <= y + - x
  elim H7; intros.
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
H8:x < x0
H9:x0 < y
a < x0 < b
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
0 <= y + - x
  split.
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
H8:x < x0
H9:x0 < y
a < x0
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
H8:x < x0
H9:x0 < y
x0 < b
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
0 <= y + - x
  elim H1; intros.
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
H8:x < x0
H9:x0 < y
H10:a <= x
H11:x <= b
a < x0
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
H8:x < x0
H9:x0 < y
x0 < b
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
0 <= y + - x
  apply Rle_lt_trans with x; assumption.
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
H8:x < x0
H9:x0 < y
x0 < b
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
0 <= y + - x
  elim H2; intros.
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
H8:x < x0
H9:x0 < y
H10:a <= y
H11:y <= b
x0 < b
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
0 <= y + - x
  apply Rlt_le_trans with y; assumption.
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
0 <= y + - x
  apply Rplus_le_reg_l with x.
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 <= derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:exists c : R,
  f y - f x = derive_pt f c (pr c) * (y - x) /\
  x < c < y
x0:R
H5:f y - f x = derive_pt f x0 (pr x0) * (y - x) /\
x < x0 < y
H6:f y + - f x = derive_pt f x0 (pr x0) * (y + - x)
H7:x < x0 < y
x + 0 <= x + (y + - x)
  rewrite Rplus_0_r; replace (x + (y + - x)) with y;
    [ left; assumption | ring ].
Qed.

(**********)
Lemma derive_increasing_interv :
  forall (a b:R) (f:R -> R) (pr:derivable f),
    a < b ->
    (forall t:R, a < t < b -> 0 < derive_pt f t (pr t)) ->
    forall x y:R, a <= x <= b -> a <= y <= b -> x < y -> f x < f y.
forall (a b : R) (f : R -> R) (pr : derivable f),
a < b ->
(forall t : R, a < t < b -> 0 < derive_pt f t (pr t)) ->
forall x y : R,
a <= x <= b -> a <= y <= b -> x < y -> f x < f y
Proof.
forall (a b : R) (f : R -> R) (pr : derivable f),
a < b ->
(forall t : R, a < t < b -> 0 < derive_pt f t (pr t)) ->
forall x y : R,
a <= x <= b -> a <= y <= b -> x < y -> f x < f y
  intros.
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 < derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
f x < f y
  generalize (derive_increasing_interv_ax a b f pr H); intro.
a, b:R
f:R -> R
pr:derivable f
H:a < b
H0:forall t : R,
a < t < b -> 0 < derive_pt f t (pr t)
x, y:R
H1:a <= x <= b
H2:a <= y <= b
H3:x < y
H4:((forall t : R,
  a < t < b -> 0 < derive_pt f t (pr t)) ->
 forall x0 y0 : R,
 a <= x0 <= b ->
 a <= y0 <= b -> x0 < y0 -> f x0 < f y0) /\
((forall t : R,
  a < t < b -> 0 <= derive_pt f t (pr t)) ->
 forall x0 y0 : R,
 a <= x0 <= b ->
 a <= y0 <= b -> x0 < y0 -> f x0 <= f y0)
f x < f y
  elim H4; intros H5 _; apply (H5 H0 x y H1 H2 H3).
Qed.

(**********)
Lemma derive_increasing_interv_var :
  forall (a b:R) (f:R -> R) (pr:derivable f),
    a < b ->
    (forall t:R, a < t < b -> 0 <= derive_pt f t (pr t)) ->
    forall x y:R, a <= x <= b -> a <= y <= b -> x < y -> f x <= f y.
forall (a b : R) (f : R -> R) (pr : derivable f),
a < b ->
(forall t : R, a < t < b -> 0 <= derive_pt f t (pr t)) ->
forall x y : R,
a <= x <= b -> a <= y <= b -> x < y -> f x <= f y
Proof.
forall (a b : R) (f : R -> R) (pr : derivable f),
a < b ->
(forall t : R, a < t < b -> 0 <= derive_pt f t (pr t)) ->
forall x y : R,
a <= x <= b -> a <= y <= b -> x < y -> f x <= f y
  intros a b f pr H H0 x y H1 H2 H3;
    generalize (derive_increasing_interv_ax a b f pr H);
      intro; elim H4; intros _ H5; apply (H5 H0 x y H1 H2 H3).
Qed.

(**********)
(**********)
Theorem IAF :
  forall (f:R -> R) (a b k:R) (pr:derivable f),
    a <= b ->
    (forall c:R, a <= c <= b -> derive_pt f c (pr c) <= k) ->
    f b - f a <= k * (b - a).
forall (f : R -> R) (a b k : R) (pr : derivable f),
a <= b ->
(forall c : R,
 a <= c <= b -> derive_pt f c (pr c) <= k) ->
f b - f a <= k * (b - a)
Proof.
forall (f : R -> R) (a b k : R) (pr : derivable f),
a <= b ->
(forall c : R,
 a <= c <= b -> derive_pt f c (pr c) <= k) ->
f b - f a <= k * (b - a)
  intros.
f:R -> R
a, b, k:R
pr:derivable f
H:a <= b
H0:forall c : R,
a <= c <= b -> derive_pt f c (pr c) <= k
f b - f a <= k * (b - a)
  destruct (total_order_T a b) as [[H1| -> ]|H1].
f:R -> R
a, b, k:R
pr:derivable f
H:a <= b
H0:forall c : R,
a <= c <= b -> derive_pt f c (pr c) <= k
H1:a < b
f b - f a <= k * (b - a)
f:R -> R
b, k:R
pr:derivable f
H0:forall c : R,
b <= c <= b -> derive_pt f c (pr c) <= k
H:b <= b
f b - f b <= k * (b - b)
f:R -> R
a, b, k:R
pr:derivable f
H:a <= b
H0:forall c : R,
a <= c <= b -> derive_pt f c (pr c) <= k
H1:a > b
f b - f a <= k * (b - a)
  pose proof (MVT_cor1 f _ _ pr H1) as (c & -> & H4).
f:R -> R
a, b, k:R
pr:derivable f
H:a <= b
H0:forall c0 : R,
a <= c0 <= b -> derive_pt f c0 (pr c0) <= k
H1:a < b
c:R
H4:a < c < b
derive_pt f c (pr c) * (b - a) <= k * (b - a)
f:R -> R
b, k:R
pr:derivable f
H0:forall c : R,
b <= c <= b -> derive_pt f c (pr c) <= k
H:b <= b
f b - f b <= k * (b - b)
f:R -> R
a, b, k:R
pr:derivable f
H:a <= b
H0:forall c : R,
a <= c <= b -> derive_pt f c (pr c) <= k
H1:a > b
f b - f a <= k * (b - a)
  do 2 rewrite <- (Rmult_comm (b - a)).
f:R -> R
a, b, k:R
pr:derivable f
H:a <= b
H0:forall c0 : R,
a <= c0 <= b -> derive_pt f c0 (pr c0) <= k
H1:a < b
c:R
H4:a < c < b
(b - a) * derive_pt f c (pr c) <= (b - a) * k
f:R -> R
b, k:R
pr:derivable f
H0:forall c : R,
b <= c <= b -> derive_pt f c (pr c) <= k
H:b <= b
f b - f b <= k * (b - b)
f:R -> R
a, b, k:R
pr:derivable f
H:a <= b
H0:forall c : R,
a <= c <= b -> derive_pt f c (pr c) <= k
H1:a > b
f b - f a <= k * (b - a)
  apply Rmult_le_compat_l.
f:R -> R
a, b, k:R
pr:derivable f
H:a <= b
H0:forall c0 : R,
a <= c0 <= b -> derive_pt f c0 (pr c0) <= k
H1:a < b
c:R
H4:a < c < b
0 <= b - a
f:R -> R
a, b, k:R
pr:derivable f
H:a <= b
H0:forall c0 : R,
a <= c0 <= b -> derive_pt f c0 (pr c0) <= k
H1:a < b
c:R
H4:a < c < b
derive_pt f c (pr c) <= k
f:R -> R
b, k:R
pr:derivable f
H0:forall c : R,
b <= c <= b -> derive_pt f c (pr c) <= k
H:b <= b
f b - f b <= k * (b - b)
f:R -> R
a, b, k:R
pr:derivable f
H:a <= b
H0:forall c : R,
a <= c <= b -> derive_pt f c (pr c) <= k
H1:a > b
f b - f a <= k * (b - a)
  apply Rplus_le_reg_l with a; rewrite Rplus_0_r.
f:R -> R
a, b, k:R
pr:derivable f
H:a <= b
H0:forall c0 : R,
a <= c0 <= b -> derive_pt f c0 (pr c0) <= k
H1:a < b
c:R
H4:a < c < b
a <= a + (b - a)
f:R -> R
a, b, k:R
pr:derivable f
H:a <= b
H0:forall c0 : R,
a <= c0 <= b -> derive_pt f c0 (pr c0) <= k
H1:a < b
c:R
H4:a < c < b
derive_pt f c (pr c) <= k
f:R -> R
b, k:R
pr:derivable f
H0:forall c : R,
b <= c <= b -> derive_pt f c (pr c) <= k
H:b <= b
f b - f b <= k * (b - b)
f:R -> R
a, b, k:R
pr:derivable f
H:a <= b
H0:forall c : R,
a <= c <= b -> derive_pt f c (pr c) <= k
H1:a > b
f b - f a <= k * (b - a)
  replace (a + (b - a)) with b; [ assumption | ring ].
f:R -> R
a, b, k:R
pr:derivable f
H:a <= b
H0:forall c0 : R,
a <= c0 <= b -> derive_pt f c0 (pr c0) <= k
H1:a < b
c:R
H4:a < c < b
derive_pt f c (pr c) <= k
f:R -> R
b, k:R
pr:derivable f
H0:forall c : R,
b <= c <= b -> derive_pt f c (pr c) <= k
H:b <= b
f b - f b <= k * (b - b)
f:R -> R
a, b, k:R
pr:derivable f
H:a <= b
H0:forall c : R,
a <= c <= b -> derive_pt f c (pr c) <= k
H1:a > b
f b - f a <= k * (b - a)
  apply H0.
f:R -> R
a, b, k:R
pr:derivable f
H:a <= b
H0:forall c0 : R,
a <= c0 <= b -> derive_pt f c0 (pr c0) <= k
H1:a < b
c:R
H4:a < c < b
a <= c <= b
f:R -> R
b, k:R
pr:derivable f
H0:forall c : R,
b <= c <= b -> derive_pt f c (pr c) <= k
H:b <= b
f b - f b <= k * (b - b)
f:R -> R
a, b, k:R
pr:derivable f
H:a <= b
H0:forall c : R,
a <= c <= b -> derive_pt f c (pr c) <= k
H1:a > b
f b - f a <= k * (b - a)
  elim H4; intros.
f:R -> R
a, b, k:R
pr:derivable f
H:a <= b
H0:forall c0 : R,
a <= c0 <= b -> derive_pt f c0 (pr c0) <= k
H1:a < b
c:R
H4:a < c < b
H2:a < c
H3:c < b
a <= c <= b
f:R -> R
b, k:R
pr:derivable f
H0:forall c : R,
b <= c <= b -> derive_pt f c (pr c) <= k
H:b <= b
f b - f b <= k * (b - b)
f:R -> R
a, b, k:R
pr:derivable f
H:a <= b
H0:forall c : R,
a <= c <= b -> derive_pt f c (pr c) <= k
H1:a > b
f b - f a <= k * (b - a)
  split; left; assumption.
f:R -> R
b, k:R
pr:derivable f
H0:forall c : R,
b <= c <= b -> derive_pt f c (pr c) <= k
H:b <= b
f b - f b <= k * (b - b)
f:R -> R
a, b, k:R
pr:derivable f
H:a <= b
H0:forall c : R,
a <= c <= b -> derive_pt f c (pr c) <= k
H1:a > b
f b - f a <= k * (b - a)
  unfold Rminus; do 2 rewrite Rplus_opp_r.
f:R -> R
b, k:R
pr:derivable f
H0:forall c : R,
b <= c <= b -> derive_pt f c (pr c) <= k
H:b <= b
0 <= k * 0
f:R -> R
a, b, k:R
pr:derivable f
H:a <= b
H0:forall c : R,
a <= c <= b -> derive_pt f c (pr c) <= k
H1:a > b
f b - f a <= k * (b - a)
  rewrite Rmult_0_r; right; reflexivity.
f:R -> R
a, b, k:R
pr:derivable f
H:a <= b
H0:forall c : R,
a <= c <= b -> derive_pt f c (pr c) <= k
H1:a > b
f b - f a <= k * (b - a)
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H H1)).
Qed.

Lemma IAF_var :
  forall (f g:R -> R) (a b:R) (pr1:derivable f) (pr2:derivable g),
    a <= b ->
    (forall c:R, a <= c <= b -> derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)) ->
    g b - g a <= f b - f a.
forall (f g : R -> R) (a b : R) (pr1 : derivable f)
  (pr2 : derivable g),
a <= b ->
(forall c : R,
 a <= c <= b ->
 derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)) ->
g b - g a <= f b - f a
Proof.
forall (f g : R -> R) (a b : R) (pr1 : derivable f)
  (pr2 : derivable g),
a <= b ->
(forall c : R,
 a <= c <= b ->
 derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)) ->
g b - g a <= f b - f a
  intros.
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
g b - g a <= f b - f a
  cut (derivable (g - f)).
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
derivable (g - f) -> g b - g a <= f b - f a
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
derivable (g - f)
  intro X.
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
X:derivable (g - f)
g b - g a <= f b - f a
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
derivable (g - f)
  cut (forall c:R, a <= c <= b -> derive_pt (g - f) c (X c) <= 0).
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
X:derivable (g - f)
(forall c : R,
 a <= c <= b -> derive_pt (g - f) c (X c) <= 0) ->
g b - g a <= f b - f a
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
X:derivable (g - f)
forall c : R,
a <= c <= b -> derive_pt (g - f) c (X c) <= 0
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
derivable (g - f)
  intro.
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
X:derivable (g - f)
H1:forall c : R,
a <= c <= b -> derive_pt (g - f) c (X c) <= 0
g b - g a <= f b - f a
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
X:derivable (g - f)
forall c : R,
a <= c <= b -> derive_pt (g - f) c (X c) <= 0
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
derivable (g - f)
  assert (H2 := IAF (g - f)%F a b 0 X H H1).
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
X:derivable (g - f)
H1:forall c : R,
a <= c <= b -> derive_pt (g - f) c (X c) <= 0
H2:(g - f)%F b - (g - f)%F a <= 0 * (b - a)
g b - g a <= f b - f a
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
X:derivable (g - f)
forall c : R,
a <= c <= b -> derive_pt (g - f) c (X c) <= 0
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
derivable (g - f)
  rewrite Rmult_0_l in H2; unfold minus_fct in H2.
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
X:derivable (g - f)
H1:forall c : R,
a <= c <= b -> derive_pt (g - f) c (X c) <= 0
H2:g b - f b - (g a - f a) <= 0
g b - g a <= f b - f a
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
X:derivable (g - f)
forall c : R,
a <= c <= b -> derive_pt (g - f) c (X c) <= 0
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
derivable (g - f)
  apply Rplus_le_reg_l with (- f b + f a).
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
X:derivable (g - f)
H1:forall c : R,
a <= c <= b -> derive_pt (g - f) c (X c) <= 0
H2:g b - f b - (g a - f a) <= 0
- f b + f a + (g b - g a) <= - f b + f a + (f b - f a)
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
X:derivable (g - f)
forall c : R,
a <= c <= b -> derive_pt (g - f) c (X c) <= 0
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
derivable (g - f)
  replace (- f b + f a + (f b - f a)) with 0; [ idtac | ring ].
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
X:derivable (g - f)
H1:forall c : R,
a <= c <= b -> derive_pt (g - f) c (X c) <= 0
H2:g b - f b - (g a - f a) <= 0
- f b + f a + (g b - g a) <= 0
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
X:derivable (g - f)
forall c : R,
a <= c <= b -> derive_pt (g - f) c (X c) <= 0
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
derivable (g - f)
  replace (- f b + f a + (g b - g a)) with (g b - f b - (g a - f a));
  [ apply H2 | ring ].
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
X:derivable (g - f)
forall c : R,
a <= c <= b -> derive_pt (g - f) c (X c) <= 0
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
derivable (g - f)
  intros.
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c0 : R,
a <= c0 <= b ->
derive_pt g c0 (pr2 c0) <=
derive_pt f c0 (pr1 c0)
X:derivable (g - f)
c:R
H1:a <= c <= b
derive_pt (g - f) c (X c) <= 0
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
derivable (g - f)
  cut
    (derive_pt (g - f) c (X c) =
      derive_pt (g - f) c (derivable_pt_minus _ _ _ (pr2 c) (pr1 c))).
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c0 : R,
a <= c0 <= b ->
derive_pt g c0 (pr2 c0) <=
derive_pt f c0 (pr1 c0)
X:derivable (g - f)
c:R
H1:a <= c <= b
derive_pt (g - f) c (X c) =
derive_pt (g - f) c
  (derivable_pt_minus g f c (pr2 c) (pr1 c)) ->
derive_pt (g - f) c (X c) <= 0
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c0 : R,
a <= c0 <= b ->
derive_pt g c0 (pr2 c0) <=
derive_pt f c0 (pr1 c0)
X:derivable (g - f)
c:R
H1:a <= c <= b
derive_pt (g - f) c (X c) =
derive_pt (g - f) c
  (derivable_pt_minus g f c (pr2 c) (pr1 c))
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
derivable (g - f)
  intro.
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c0 : R,
a <= c0 <= b ->
derive_pt g c0 (pr2 c0) <=
derive_pt f c0 (pr1 c0)
X:derivable (g - f)
c:R
H1:a <= c <= b
H2:derive_pt (g - f) c (X c) =
derive_pt (g - f) c
  (derivable_pt_minus g f c (pr2 c) (pr1 c))
derive_pt (g - f) c (X c) <= 0
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c0 : R,
a <= c0 <= b ->
derive_pt g c0 (pr2 c0) <=
derive_pt f c0 (pr1 c0)
X:derivable (g - f)
c:R
H1:a <= c <= b
derive_pt (g - f) c (X c) =
derive_pt (g - f) c
  (derivable_pt_minus g f c (pr2 c) (pr1 c))
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
derivable (g - f)
  rewrite H2.
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c0 : R,
a <= c0 <= b ->
derive_pt g c0 (pr2 c0) <=
derive_pt f c0 (pr1 c0)
X:derivable (g - f)
c:R
H1:a <= c <= b
H2:derive_pt (g - f) c (X c) =
derive_pt (g - f) c
  (derivable_pt_minus g f c (pr2 c) (pr1 c))
derive_pt (g - f) c
  (derivable_pt_minus g f c (pr2 c) (pr1 c)) <= 0
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c0 : R,
a <= c0 <= b ->
derive_pt g c0 (pr2 c0) <=
derive_pt f c0 (pr1 c0)
X:derivable (g - f)
c:R
H1:a <= c <= b
derive_pt (g - f) c (X c) =
derive_pt (g - f) c
  (derivable_pt_minus g f c (pr2 c) (pr1 c))
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
derivable (g - f)
  rewrite derive_pt_minus.
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c0 : R,
a <= c0 <= b ->
derive_pt g c0 (pr2 c0) <=
derive_pt f c0 (pr1 c0)
X:derivable (g - f)
c:R
H1:a <= c <= b
H2:derive_pt (g - f) c (X c) =
derive_pt (g - f) c
  (derivable_pt_minus g f c (pr2 c) (pr1 c))
derive_pt g c (pr2 c) - derive_pt f c (pr1 c) <= 0
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c0 : R,
a <= c0 <= b ->
derive_pt g c0 (pr2 c0) <=
derive_pt f c0 (pr1 c0)
X:derivable (g - f)
c:R
H1:a <= c <= b
derive_pt (g - f) c (X c) =
derive_pt (g - f) c
  (derivable_pt_minus g f c (pr2 c) (pr1 c))
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
derivable (g - f)
  apply Rplus_le_reg_l with (derive_pt f c (pr1 c)).
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c0 : R,
a <= c0 <= b ->
derive_pt g c0 (pr2 c0) <=
derive_pt f c0 (pr1 c0)
X:derivable (g - f)
c:R
H1:a <= c <= b
H2:derive_pt (g - f) c (X c) =
derive_pt (g - f) c
  (derivable_pt_minus g f c (pr2 c) (pr1 c))
derive_pt f c (pr1 c) +
(derive_pt g c (pr2 c) - derive_pt f c (pr1 c)) <=
derive_pt f c (pr1 c) + 0
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c0 : R,
a <= c0 <= b ->
derive_pt g c0 (pr2 c0) <=
derive_pt f c0 (pr1 c0)
X:derivable (g - f)
c:R
H1:a <= c <= b
derive_pt (g - f) c (X c) =
derive_pt (g - f) c
  (derivable_pt_minus g f c (pr2 c) (pr1 c))
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
derivable (g - f)
  rewrite Rplus_0_r.
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c0 : R,
a <= c0 <= b ->
derive_pt g c0 (pr2 c0) <=
derive_pt f c0 (pr1 c0)
X:derivable (g - f)
c:R
H1:a <= c <= b
H2:derive_pt (g - f) c (X c) =
derive_pt (g - f) c
  (derivable_pt_minus g f c (pr2 c) (pr1 c))
derive_pt f c (pr1 c) +
(derive_pt g c (pr2 c) - derive_pt f c (pr1 c)) <=
derive_pt f c (pr1 c)
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c0 : R,
a <= c0 <= b ->
derive_pt g c0 (pr2 c0) <=
derive_pt f c0 (pr1 c0)
X:derivable (g - f)
c:R
H1:a <= c <= b
derive_pt (g - f) c (X c) =
derive_pt (g - f) c
  (derivable_pt_minus g f c (pr2 c) (pr1 c))
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
derivable (g - f)
  replace
  (derive_pt f c (pr1 c) + (derive_pt g c (pr2 c) - derive_pt f c (pr1 c)))
    with (derive_pt g c (pr2 c)); [ idtac | ring ].
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c0 : R,
a <= c0 <= b ->
derive_pt g c0 (pr2 c0) <=
derive_pt f c0 (pr1 c0)
X:derivable (g - f)
c:R
H1:a <= c <= b
H2:derive_pt (g - f) c (X c) =
derive_pt (g - f) c
  (derivable_pt_minus g f c (pr2 c) (pr1 c))
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c0 : R,
a <= c0 <= b ->
derive_pt g c0 (pr2 c0) <=
derive_pt f c0 (pr1 c0)
X:derivable (g - f)
c:R
H1:a <= c <= b
derive_pt (g - f) c (X c) =
derive_pt (g - f) c
  (derivable_pt_minus g f c (pr2 c) (pr1 c))
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
derivable (g - f)
  apply H0; assumption.
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c0 : R,
a <= c0 <= b ->
derive_pt g c0 (pr2 c0) <=
derive_pt f c0 (pr1 c0)
X:derivable (g - f)
c:R
H1:a <= c <= b
derive_pt (g - f) c (X c) =
derive_pt (g - f) c
  (derivable_pt_minus g f c (pr2 c) (pr1 c))
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
derivable (g - f)
  apply pr_nu.
f, g:R -> R
a, b:R
pr1:derivable f
pr2:derivable g
H:a <= b
H0:forall c : R,
a <= c <= b ->
derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)
derivable (g - f)
  apply derivable_minus; assumption.
Qed.

(* If f has a null derivative in ]a,b[ and is continue in [a,b], *)
(* then f is constant on [a,b] *)
Lemma null_derivative_loc :
  forall (f:R -> R) (a b:R) (pr:forall x:R, a < x < b -> derivable_pt f x),
    (forall x:R, a <= x <= b -> continuity_pt f x) ->
    (forall (x:R) (P:a < x < b), derive_pt f x (pr x P) = 0) ->
    constant_D_eq f (fun x:R => a <= x <= b) (f a).
forall (f : R -> R) (a b : R)
  (pr : forall x : R, a < x < b -> derivable_pt f x),
(forall x : R, a <= x <= b -> continuity_pt f x) ->
(forall (x : R) (P : a < x < b),
 derive_pt f x (pr x P) = 0) ->
constant_D_eq f (fun x : R => a <= x <= b) (f a)
Proof.
forall (f : R -> R) (a b : R)
  (pr : forall x : R, a < x < b -> derivable_pt f x),
(forall x : R, a <= x <= b -> continuity_pt f x) ->
(forall (x : R) (P : a < x < b),
 derive_pt f x (pr x P) = 0) ->
constant_D_eq f (fun x : R => a <= x <= b) (f a)
  intros; unfold constant_D_eq; intros; destruct (total_order_T a b) as [[Hlt|Heq]|Hgt].
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hlt:a < b
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Heq:a = b
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hgt:a > b
f x = f a
  assert (H2 : forall y:R, a < y < x -> derivable_pt id y).
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hlt:a < b
forall y : R, a < y < x -> derivable_pt id y
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Heq:a = b
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hgt:a > b
f x = f a
  intros; apply derivable_pt_id.
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Heq:a = b
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hgt:a > b
f x = f a
  assert (H3 : forall y:R, a <= y <= x -> continuity_pt id y).
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
forall y : R, a <= y <= x -> continuity_pt id y
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Heq:a = b
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hgt:a > b
f x = f a
  intros; apply derivable_continuous; apply derivable_id.
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Heq:a = b
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hgt:a > b
f x = f a
  assert (H4 : forall y:R, a < y < x -> derivable_pt f y).
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
forall y : R, a < y < x -> derivable_pt f y
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Heq:a = b
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hgt:a > b
f x = f a
  intros; apply pr; elim H4; intros; split.
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hlt:a < b
H2:forall y0 : R, a < y0 < x -> derivable_pt id y0
H3:forall y0 : R,
a <= y0 <= x -> continuity_pt id y0
y:R
H4:a < y < x
H5:a < y
H6:y < x
a < y
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hlt:a < b
H2:forall y0 : R, a < y0 < x -> derivable_pt id y0
H3:forall y0 : R,
a <= y0 <= x -> continuity_pt id y0
y:R
H4:a < y < x
H5:a < y
H6:y < x
y < b
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Heq:a = b
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hgt:a > b
f x = f a
  assumption.
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hlt:a < b
H2:forall y0 : R, a < y0 < x -> derivable_pt id y0
H3:forall y0 : R,
a <= y0 <= x -> continuity_pt id y0
y:R
H4:a < y < x
H5:a < y
H6:y < x
y < b
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Heq:a = b
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hgt:a > b
f x = f a
  elim H1; intros; apply Rlt_le_trans with x; assumption.
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Heq:a = b
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hgt:a > b
f x = f a
  assert (H5 : forall y:R, a <= y <= x -> continuity_pt f y).
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
forall y : R, a <= y <= x -> continuity_pt f y
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
H5:forall y : R, a <= y <= x -> continuity_pt f y
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Heq:a = b
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hgt:a > b
f x = f a
  intros; apply H; elim H5; intros; split.
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hlt:a < b
H2:forall y0 : R, a < y0 < x -> derivable_pt id y0
H3:forall y0 : R,
a <= y0 <= x -> continuity_pt id y0
H4:forall y0 : R, a < y0 < x -> derivable_pt f y0
y:R
H5:a <= y <= x
H6:a <= y
H7:y <= x
a <= y
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hlt:a < b
H2:forall y0 : R, a < y0 < x -> derivable_pt id y0
H3:forall y0 : R,
a <= y0 <= x -> continuity_pt id y0
H4:forall y0 : R, a < y0 < x -> derivable_pt f y0
y:R
H5:a <= y <= x
H6:a <= y
H7:y <= x
y <= b
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
H5:forall y : R, a <= y <= x -> continuity_pt f y
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Heq:a = b
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hgt:a > b
f x = f a
  assumption.
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hlt:a < b
H2:forall y0 : R, a < y0 < x -> derivable_pt id y0
H3:forall y0 : R,
a <= y0 <= x -> continuity_pt id y0
H4:forall y0 : R, a < y0 < x -> derivable_pt f y0
y:R
H5:a <= y <= x
H6:a <= y
H7:y <= x
y <= b
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
H5:forall y : R, a <= y <= x -> continuity_pt f y
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Heq:a = b
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hgt:a > b
f x = f a
  elim H1; intros; apply Rle_trans with x; assumption.
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
H5:forall y : R, a <= y <= x -> continuity_pt f y
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Heq:a = b
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hgt:a > b
f x = f a
  elim H1; clear H1; intros; elim H1; clear H1; intro.
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
H5:forall y : R, a <= y <= x -> continuity_pt f y
H6:x <= b
H1:a < x
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
H5:forall y : R, a <= y <= x -> continuity_pt f y
H6:x <= b
H1:a = x
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Heq:a = b
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hgt:a > b
f x = f a
  assert (H7 := MVT f id a x H4 H2 H1 H5 H3).
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
H5:forall y : R, a <= y <= x -> continuity_pt f y
H6:x <= b
H1:a < x
H7:exists (c : R) (P : a < c < x),
  (id x - id a) * derive_pt f c (H4 c P) =
  (f x - f a) * derive_pt id c (H2 c P)
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
H5:forall y : R, a <= y <= x -> continuity_pt f y
H6:x <= b
H1:a = x
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Heq:a = b
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hgt:a > b
f x = f a
  destruct H7 as (c & P & H9).
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P0 : a < x0 < b),
derive_pt f x0 (pr x0 P0) = 0
x:R
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
H5:forall y : R, a <= y <= x -> continuity_pt f y
H6:x <= b
H1:a < x
c:R
P:a < c < x
H9:(id x - id a) * derive_pt f c (H4 c P) =
(f x - f a) * derive_pt id c (H2 c P)
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
H5:forall y : R, a <= y <= x -> continuity_pt f y
H6:x <= b
H1:a = x
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Heq:a = b
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hgt:a > b
f x = f a
  assert (H10 : a < c < b).
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P0 : a < x0 < b),
derive_pt f x0 (pr x0 P0) = 0
x:R
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
H5:forall y : R, a <= y <= x -> continuity_pt f y
H6:x <= b
H1:a < x
c:R
P:a < c < x
H9:(id x - id a) * derive_pt f c (H4 c P) =
(f x - f a) * derive_pt id c (H2 c P)
a < c < b
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P0 : a < x0 < b),
derive_pt f x0 (pr x0 P0) = 0
x:R
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
H5:forall y : R, a <= y <= x -> continuity_pt f y
H6:x <= b
H1:a < x
c:R
P:a < c < x
H9:(id x - id a) * derive_pt f c (H4 c P) =
(f x - f a) * derive_pt id c (H2 c P)
H10:a < c < b
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
H5:forall y : R, a <= y <= x -> continuity_pt f y
H6:x <= b
H1:a = x
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Heq:a = b
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hgt:a > b
f x = f a
  split.
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P0 : a < x0 < b),
derive_pt f x0 (pr x0 P0) = 0
x:R
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
H5:forall y : R, a <= y <= x -> continuity_pt f y
H6:x <= b
H1:a < x
c:R
P:a < c < x
H9:(id x - id a) * derive_pt f c (H4 c P) =
(f x - f a) * derive_pt id c (H2 c P)
a < c
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P0 : a < x0 < b),
derive_pt f x0 (pr x0 P0) = 0
x:R
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
H5:forall y : R, a <= y <= x -> continuity_pt f y
H6:x <= b
H1:a < x
c:R
P:a < c < x
H9:(id x - id a) * derive_pt f c (H4 c P) =
(f x - f a) * derive_pt id c (H2 c P)
c < b
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P0 : a < x0 < b),
derive_pt f x0 (pr x0 P0) = 0
x:R
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
H5:forall y : R, a <= y <= x -> continuity_pt f y
H6:x <= b
H1:a < x
c:R
P:a < c < x
H9:(id x - id a) * derive_pt f c (H4 c P) =
(f x - f a) * derive_pt id c (H2 c P)
H10:a < c < b
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
H5:forall y : R, a <= y <= x -> continuity_pt f y
H6:x <= b
H1:a = x
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Heq:a = b
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hgt:a > b
f x = f a
  apply P.
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P0 : a < x0 < b),
derive_pt f x0 (pr x0 P0) = 0
x:R
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
H5:forall y : R, a <= y <= x -> continuity_pt f y
H6:x <= b
H1:a < x
c:R
P:a < c < x
H9:(id x - id a) * derive_pt f c (H4 c P) =
(f x - f a) * derive_pt id c (H2 c P)
c < b
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P0 : a < x0 < b),
derive_pt f x0 (pr x0 P0) = 0
x:R
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
H5:forall y : R, a <= y <= x -> continuity_pt f y
H6:x <= b
H1:a < x
c:R
P:a < c < x
H9:(id x - id a) * derive_pt f c (H4 c P) =
(f x - f a) * derive_pt id c (H2 c P)
H10:a < c < b
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
H5:forall y : R, a <= y <= x -> continuity_pt f y
H6:x <= b
H1:a = x
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Heq:a = b
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hgt:a > b
f x = f a
  apply Rlt_le_trans with x; [apply P|assumption].
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P0 : a < x0 < b),
derive_pt f x0 (pr x0 P0) = 0
x:R
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
H5:forall y : R, a <= y <= x -> continuity_pt f y
H6:x <= b
H1:a < x
c:R
P:a < c < x
H9:(id x - id a) * derive_pt f c (H4 c P) =
(f x - f a) * derive_pt id c (H2 c P)
H10:a < c < b
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
H5:forall y : R, a <= y <= x -> continuity_pt f y
H6:x <= b
H1:a = x
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Heq:a = b
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hgt:a > b
f x = f a
  assert (H11 : derive_pt f c (H4 c P) = 0).
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P0 : a < x0 < b),
derive_pt f x0 (pr x0 P0) = 0
x:R
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
H5:forall y : R, a <= y <= x -> continuity_pt f y
H6:x <= b
H1:a < x
c:R
P:a < c < x
H9:(id x - id a) * derive_pt f c (H4 c P) =
(f x - f a) * derive_pt id c (H2 c P)
H10:a < c < b
derive_pt f c (H4 c P) = 0
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P0 : a < x0 < b),
derive_pt f x0 (pr x0 P0) = 0
x:R
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
H5:forall y : R, a <= y <= x -> continuity_pt f y
H6:x <= b
H1:a < x
c:R
P:a < c < x
H9:(id x - id a) * derive_pt f c (H4 c P) =
(f x - f a) * derive_pt id c (H2 c P)
H10:a < c < b
H11:derive_pt f c (H4 c P) = 0
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
H5:forall y : R, a <= y <= x -> continuity_pt f y
H6:x <= b
H1:a = x
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Heq:a = b
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hgt:a > b
f x = f a
  replace (derive_pt f c (H4 c P)) with (derive_pt f c (pr c H10));
  [ apply H0 | apply pr_nu ].
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P0 : a < x0 < b),
derive_pt f x0 (pr x0 P0) = 0
x:R
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
H5:forall y : R, a <= y <= x -> continuity_pt f y
H6:x <= b
H1:a < x
c:R
P:a < c < x
H9:(id x - id a) * derive_pt f c (H4 c P) =
(f x - f a) * derive_pt id c (H2 c P)
H10:a < c < b
H11:derive_pt f c (H4 c P) = 0
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
H5:forall y : R, a <= y <= x -> continuity_pt f y
H6:x <= b
H1:a = x
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Heq:a = b
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hgt:a > b
f x = f a
  assert (H12 : derive_pt id c (H2 c P) = 1).
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P0 : a < x0 < b),
derive_pt f x0 (pr x0 P0) = 0
x:R
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
H5:forall y : R, a <= y <= x -> continuity_pt f y
H6:x <= b
H1:a < x
c:R
P:a < c < x
H9:(id x - id a) * derive_pt f c (H4 c P) =
(f x - f a) * derive_pt id c (H2 c P)
H10:a < c < b
H11:derive_pt f c (H4 c P) = 0
derive_pt id c (H2 c P) = 1
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P0 : a < x0 < b),
derive_pt f x0 (pr x0 P0) = 0
x:R
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
H5:forall y : R, a <= y <= x -> continuity_pt f y
H6:x <= b
H1:a < x
c:R
P:a < c < x
H9:(id x - id a) * derive_pt f c (H4 c P) =
(f x - f a) * derive_pt id c (H2 c P)
H10:a < c < b
H11:derive_pt f c (H4 c P) = 0
H12:derive_pt id c (H2 c P) = 1
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
H5:forall y : R, a <= y <= x -> continuity_pt f y
H6:x <= b
H1:a = x
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Heq:a = b
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hgt:a > b
f x = f a
  apply derive_pt_eq_0; apply derivable_pt_lim_id.
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P0 : a < x0 < b),
derive_pt f x0 (pr x0 P0) = 0
x:R
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
H5:forall y : R, a <= y <= x -> continuity_pt f y
H6:x <= b
H1:a < x
c:R
P:a < c < x
H9:(id x - id a) * derive_pt f c (H4 c P) =
(f x - f a) * derive_pt id c (H2 c P)
H10:a < c < b
H11:derive_pt f c (H4 c P) = 0
H12:derive_pt id c (H2 c P) = 1
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
H5:forall y : R, a <= y <= x -> continuity_pt f y
H6:x <= b
H1:a = x
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Heq:a = b
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hgt:a > b
f x = f a
  rewrite H11 in H9; rewrite H12 in H9; rewrite Rmult_0_r in H9;
    rewrite Rmult_1_r in H9; apply Rminus_diag_uniq; symmetry ;
      assumption.
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
Hlt:a < b
H2:forall y : R, a < y < x -> derivable_pt id y
H3:forall y : R, a <= y <= x -> continuity_pt id y
H4:forall y : R, a < y < x -> derivable_pt f y
H5:forall y : R, a <= y <= x -> continuity_pt f y
H6:x <= b
H1:a = x
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Heq:a = b
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hgt:a > b
f x = f a
  rewrite H1; reflexivity.
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Heq:a = b
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hgt:a > b
f x = f a
  assert (H2 : x = a).
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Heq:a = b
x = a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Heq:a = b
H2:x = a
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hgt:a > b
f x = f a
  rewrite <- Heq in H1; elim H1; intros; apply Rle_antisym; assumption.
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Heq:a = b
H2:x = a
f x = f a
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hgt:a > b
f x = f a
  rewrite H2; reflexivity.
f:R -> R
a, b:R
pr:forall x0 : R, a < x0 < b -> derivable_pt f x0
H:forall x0 : R, a <= x0 <= b -> continuity_pt f x0
H0:forall (x0 : R) (P : a < x0 < b),
derive_pt f x0 (pr x0 P) = 0
x:R
H1:a <= x <= b
Hgt:a > b
f x = f a
  elim H1; intros;
    elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ (Rle_trans _ _ _ H2 H3) Hgt)).
Qed.

(* Unicity of the antiderivative *)
Lemma antiderivative_Ucte :
  forall (f g1 g2:R -> R) (a b:R),
    antiderivative f g1 a b ->
    antiderivative f g2 a b ->
    exists c : R, (forall x:R, a <= x <= b -> g1 x = g2 x + c).
forall (f g1 g2 : R -> R) (a b : R),
antiderivative f g1 a b ->
antiderivative f g2 a b ->
exists c : R,
  forall x : R, a <= x <= b -> g1 x = g2 x + c
Proof.
forall (f g1 g2 : R -> R) (a b : R),
antiderivative f g1 a b ->
antiderivative f g2 a b ->
exists c : R,
  forall x : R, a <= x <= b -> g1 x = g2 x + c
  unfold antiderivative; intros; elim H; clear H; intros; elim H0;
    clear H0; intros H0 _; exists (g1 a - g2 a); intros;
      assert (H3 : forall x:R, a <= x <= b -> derivable_pt g1 x).
f, g1, g2:R -> R
a, b:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g1 x0,
  f x0 = derive_pt g1 x0 pr
H1:a <= b
H0:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g2 x0,
  f x0 = derive_pt g2 x0 pr
x:R
H2:a <= x <= b
forall x0 : R, a <= x0 <= b -> derivable_pt g1 x0
f, g1, g2:R -> R
a, b:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g1 x0,
  f x0 = derive_pt g1 x0 pr
H1:a <= b
H0:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g2 x0,
  f x0 = derive_pt g2 x0 pr
x:R
H2:a <= x <= b
H3:forall x0 : R, a <= x0 <= b -> derivable_pt g1 x0
g1 x = g2 x + (g1 a - g2 a)
  intros; unfold derivable_pt; exists (f x0); elim (H x0 H3);
    intros; eapply derive_pt_eq_1; symmetry ;
      apply H4.
f, g1, g2:R -> R
a, b:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g1 x0,
  f x0 = derive_pt g1 x0 pr
H1:a <= b
H0:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g2 x0,
  f x0 = derive_pt g2 x0 pr
x:R
H2:a <= x <= b
H3:forall x0 : R, a <= x0 <= b -> derivable_pt g1 x0
g1 x = g2 x + (g1 a - g2 a)
  assert (H4 : forall x:R, a <= x <= b -> derivable_pt g2 x).
f, g1, g2:R -> R
a, b:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g1 x0,
  f x0 = derive_pt g1 x0 pr
H1:a <= b
H0:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g2 x0,
  f x0 = derive_pt g2 x0 pr
x:R
H2:a <= x <= b
H3:forall x0 : R, a <= x0 <= b -> derivable_pt g1 x0
forall x0 : R, a <= x0 <= b -> derivable_pt g2 x0
f, g1, g2:R -> R
a, b:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g1 x0,
  f x0 = derive_pt g1 x0 pr
H1:a <= b
H0:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g2 x0,
  f x0 = derive_pt g2 x0 pr
x:R
H2:a <= x <= b
H3:forall x0 : R, a <= x0 <= b -> derivable_pt g1 x0
H4:forall x0 : R, a <= x0 <= b -> derivable_pt g2 x0
g1 x = g2 x + (g1 a - g2 a)
  intros; unfold derivable_pt; exists (f x0);
    elim (H0 x0 H4); intros; eapply derive_pt_eq_1; symmetry ;
      apply H5.
f, g1, g2:R -> R
a, b:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g1 x0,
  f x0 = derive_pt g1 x0 pr
H1:a <= b
H0:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g2 x0,
  f x0 = derive_pt g2 x0 pr
x:R
H2:a <= x <= b
H3:forall x0 : R, a <= x0 <= b -> derivable_pt g1 x0
H4:forall x0 : R, a <= x0 <= b -> derivable_pt g2 x0
g1 x = g2 x + (g1 a - g2 a)
  assert (H5 : forall x:R, a < x < b -> derivable_pt (g1 - g2) x).
f, g1, g2:R -> R
a, b:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g1 x0,
  f x0 = derive_pt g1 x0 pr
H1:a <= b
H0:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g2 x0,
  f x0 = derive_pt g2 x0 pr
x:R
H2:a <= x <= b
H3:forall x0 : R, a <= x0 <= b -> derivable_pt g1 x0
H4:forall x0 : R, a <= x0 <= b -> derivable_pt g2 x0
forall x0 : R, a < x0 < b -> derivable_pt (g1 - g2) x0
f, g1, g2:R -> R
a, b:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g1 x0,
  f x0 = derive_pt g1 x0 pr
H1:a <= b
H0:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g2 x0,
  f x0 = derive_pt g2 x0 pr
x:R
H2:a <= x <= b
H3:forall x0 : R, a <= x0 <= b -> derivable_pt g1 x0
H4:forall x0 : R, a <= x0 <= b -> derivable_pt g2 x0
H5:forall x0 : R,
a < x0 < b -> derivable_pt (g1 - g2) x0
g1 x = g2 x + (g1 a - g2 a)
  intros; elim H5; intros; apply derivable_pt_minus;
    [ apply H3; split; left; assumption | apply H4; split; left; assumption ].
f, g1, g2:R -> R
a, b:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g1 x0,
  f x0 = derive_pt g1 x0 pr
H1:a <= b
H0:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g2 x0,
  f x0 = derive_pt g2 x0 pr
x:R
H2:a <= x <= b
H3:forall x0 : R, a <= x0 <= b -> derivable_pt g1 x0
H4:forall x0 : R, a <= x0 <= b -> derivable_pt g2 x0
H5:forall x0 : R,
a < x0 < b -> derivable_pt (g1 - g2) x0
g1 x = g2 x + (g1 a - g2 a)
  assert (H6 : forall x:R, a <= x <= b -> continuity_pt (g1 - g2) x).
f, g1, g2:R -> R
a, b:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g1 x0,
  f x0 = derive_pt g1 x0 pr
H1:a <= b
H0:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g2 x0,
  f x0 = derive_pt g2 x0 pr
x:R
H2:a <= x <= b
H3:forall x0 : R, a <= x0 <= b -> derivable_pt g1 x0
H4:forall x0 : R, a <= x0 <= b -> derivable_pt g2 x0
H5:forall x0 : R,
a < x0 < b -> derivable_pt (g1 - g2) x0
forall x0 : R,
a <= x0 <= b -> continuity_pt (g1 - g2) x0
f, g1, g2:R -> R
a, b:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g1 x0,
  f x0 = derive_pt g1 x0 pr
H1:a <= b
H0:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g2 x0,
  f x0 = derive_pt g2 x0 pr
x:R
H2:a <= x <= b
H3:forall x0 : R, a <= x0 <= b -> derivable_pt g1 x0
H4:forall x0 : R, a <= x0 <= b -> derivable_pt g2 x0
H5:forall x0 : R,
a < x0 < b -> derivable_pt (g1 - g2) x0
H6:forall x0 : R,
a <= x0 <= b -> continuity_pt (g1 - g2) x0
g1 x = g2 x + (g1 a - g2 a)
  intros; apply derivable_continuous_pt; apply derivable_pt_minus;
    [ apply H3 | apply H4 ]; assumption.
f, g1, g2:R -> R
a, b:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g1 x0,
  f x0 = derive_pt g1 x0 pr
H1:a <= b
H0:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g2 x0,
  f x0 = derive_pt g2 x0 pr
x:R
H2:a <= x <= b
H3:forall x0 : R, a <= x0 <= b -> derivable_pt g1 x0
H4:forall x0 : R, a <= x0 <= b -> derivable_pt g2 x0
H5:forall x0 : R,
a < x0 < b -> derivable_pt (g1 - g2) x0
H6:forall x0 : R,
a <= x0 <= b -> continuity_pt (g1 - g2) x0
g1 x = g2 x + (g1 a - g2 a)
  assert (H7 : forall (x:R) (P:a < x < b), derive_pt (g1 - g2) x (H5 x P) = 0).
f, g1, g2:R -> R
a, b:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g1 x0,
  f x0 = derive_pt g1 x0 pr
H1:a <= b
H0:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g2 x0,
  f x0 = derive_pt g2 x0 pr
x:R
H2:a <= x <= b
H3:forall x0 : R, a <= x0 <= b -> derivable_pt g1 x0
H4:forall x0 : R, a <= x0 <= b -> derivable_pt g2 x0
H5:forall x0 : R,
a < x0 < b -> derivable_pt (g1 - g2) x0
H6:forall x0 : R,
a <= x0 <= b -> continuity_pt (g1 - g2) x0
forall (x0 : R) (P : a < x0 < b),
derive_pt (g1 - g2) x0 (H5 x0 P) = 0
f, g1, g2:R -> R
a, b:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g1 x0,
  f x0 = derive_pt g1 x0 pr
H1:a <= b
H0:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g2 x0,
  f x0 = derive_pt g2 x0 pr
x:R
H2:a <= x <= b
H3:forall x0 : R, a <= x0 <= b -> derivable_pt g1 x0
H4:forall x0 : R, a <= x0 <= b -> derivable_pt g2 x0
H5:forall x0 : R,
a < x0 < b -> derivable_pt (g1 - g2) x0
H6:forall x0 : R,
a <= x0 <= b -> continuity_pt (g1 - g2) x0
H7:forall (x0 : R) (P : a < x0 < b),
derive_pt (g1 - g2) x0 (H5 x0 P) = 0
g1 x = g2 x + (g1 a - g2 a)
  intros; elim P; intros; apply derive_pt_eq_0; replace 0 with (f x0 - f x0);
    [ idtac | ring ].
f, g1, g2:R -> R
a, b:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt g1 x1,
  f x1 = derive_pt g1 x1 pr
H1:a <= b
H0:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt g2 x1,
  f x1 = derive_pt g2 x1 pr
x:R
H2:a <= x <= b
H3:forall x1 : R, a <= x1 <= b -> derivable_pt g1 x1
H4:forall x1 : R, a <= x1 <= b -> derivable_pt g2 x1
H5:forall x1 : R,
a < x1 < b -> derivable_pt (g1 - g2) x1
H6:forall x1 : R,
a <= x1 <= b -> continuity_pt (g1 - g2) x1
x0:R
P:a < x0 < b
H7:a < x0
H8:x0 < b
derivable_pt_lim (g1 - g2) x0 (f x0 - f x0)
f, g1, g2:R -> R
a, b:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g1 x0,
  f x0 = derive_pt g1 x0 pr
H1:a <= b
H0:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g2 x0,
  f x0 = derive_pt g2 x0 pr
x:R
H2:a <= x <= b
H3:forall x0 : R, a <= x0 <= b -> derivable_pt g1 x0
H4:forall x0 : R, a <= x0 <= b -> derivable_pt g2 x0
H5:forall x0 : R,
a < x0 < b -> derivable_pt (g1 - g2) x0
H6:forall x0 : R,
a <= x0 <= b -> continuity_pt (g1 - g2) x0
H7:forall (x0 : R) (P : a < x0 < b),
derive_pt (g1 - g2) x0 (H5 x0 P) = 0
g1 x = g2 x + (g1 a - g2 a)
  assert (H9 : a <= x0 <= b).
f, g1, g2:R -> R
a, b:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt g1 x1,
  f x1 = derive_pt g1 x1 pr
H1:a <= b
H0:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt g2 x1,
  f x1 = derive_pt g2 x1 pr
x:R
H2:a <= x <= b
H3:forall x1 : R, a <= x1 <= b -> derivable_pt g1 x1
H4:forall x1 : R, a <= x1 <= b -> derivable_pt g2 x1
H5:forall x1 : R,
a < x1 < b -> derivable_pt (g1 - g2) x1
H6:forall x1 : R,
a <= x1 <= b -> continuity_pt (g1 - g2) x1
x0:R
P:a < x0 < b
H7:a < x0
H8:x0 < b
a <= x0 <= b
f, g1, g2:R -> R
a, b:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt g1 x1,
  f x1 = derive_pt g1 x1 pr
H1:a <= b
H0:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt g2 x1,
  f x1 = derive_pt g2 x1 pr
x:R
H2:a <= x <= b
H3:forall x1 : R, a <= x1 <= b -> derivable_pt g1 x1
H4:forall x1 : R, a <= x1 <= b -> derivable_pt g2 x1
H5:forall x1 : R,
a < x1 < b -> derivable_pt (g1 - g2) x1
H6:forall x1 : R,
a <= x1 <= b -> continuity_pt (g1 - g2) x1
x0:R
P:a < x0 < b
H7:a < x0
H8:x0 < b
H9:a <= x0 <= b
derivable_pt_lim (g1 - g2) x0 (f x0 - f x0)
f, g1, g2:R -> R
a, b:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g1 x0,
  f x0 = derive_pt g1 x0 pr
H1:a <= b
H0:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g2 x0,
  f x0 = derive_pt g2 x0 pr
x:R
H2:a <= x <= b
H3:forall x0 : R, a <= x0 <= b -> derivable_pt g1 x0
H4:forall x0 : R, a <= x0 <= b -> derivable_pt g2 x0
H5:forall x0 : R,
a < x0 < b -> derivable_pt (g1 - g2) x0
H6:forall x0 : R,
a <= x0 <= b -> continuity_pt (g1 - g2) x0
H7:forall (x0 : R) (P : a < x0 < b),
derive_pt (g1 - g2) x0 (H5 x0 P) = 0
g1 x = g2 x + (g1 a - g2 a)
  split; left; assumption.
f, g1, g2:R -> R
a, b:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt g1 x1,
  f x1 = derive_pt g1 x1 pr
H1:a <= b
H0:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt g2 x1,
  f x1 = derive_pt g2 x1 pr
x:R
H2:a <= x <= b
H3:forall x1 : R, a <= x1 <= b -> derivable_pt g1 x1
H4:forall x1 : R, a <= x1 <= b -> derivable_pt g2 x1
H5:forall x1 : R,
a < x1 < b -> derivable_pt (g1 - g2) x1
H6:forall x1 : R,
a <= x1 <= b -> continuity_pt (g1 - g2) x1
x0:R
P:a < x0 < b
H7:a < x0
H8:x0 < b
H9:a <= x0 <= b
derivable_pt_lim (g1 - g2) x0 (f x0 - f x0)
f, g1, g2:R -> R
a, b:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g1 x0,
  f x0 = derive_pt g1 x0 pr
H1:a <= b
H0:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g2 x0,
  f x0 = derive_pt g2 x0 pr
x:R
H2:a <= x <= b
H3:forall x0 : R, a <= x0 <= b -> derivable_pt g1 x0
H4:forall x0 : R, a <= x0 <= b -> derivable_pt g2 x0
H5:forall x0 : R,
a < x0 < b -> derivable_pt (g1 - g2) x0
H6:forall x0 : R,
a <= x0 <= b -> continuity_pt (g1 - g2) x0
H7:forall (x0 : R) (P : a < x0 < b),
derive_pt (g1 - g2) x0 (H5 x0 P) = 0
g1 x = g2 x + (g1 a - g2 a)
  apply derivable_pt_lim_minus; [ elim (H _ H9) | elim (H0 _ H9) ]; intros;
    eapply derive_pt_eq_1; symmetry ; apply H10.
f, g1, g2:R -> R
a, b:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g1 x0,
  f x0 = derive_pt g1 x0 pr
H1:a <= b
H0:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt g2 x0,
  f x0 = derive_pt g2 x0 pr
x:R
H2:a <= x <= b
H3:forall x0 : R, a <= x0 <= b -> derivable_pt g1 x0
H4:forall x0 : R, a <= x0 <= b -> derivable_pt g2 x0
H5:forall x0 : R,
a < x0 < b -> derivable_pt (g1 - g2) x0
H6:forall x0 : R,
a <= x0 <= b -> continuity_pt (g1 - g2) x0
H7:forall (x0 : R) (P : a < x0 < b),
derive_pt (g1 - g2) x0 (H5 x0 P) = 0
g1 x = g2 x + (g1 a - g2 a)
  assert (H8 := null_derivative_loc (g1 - g2)%F a b H5 H6 H7);
    unfold constant_D_eq in H8; assert (H9 := H8 _ H2);
      unfold minus_fct in H9; rewrite <- H9; ring.
Qed.

(* A variant of MVT using absolute values. *)
Lemma MVT_abs : 
  forall (f f' : R -> R) (a b : R),
  (forall c : R, Rmin a b <= c <= Rmax a b ->
      derivable_pt_lim f c (f' c)) ->
  exists c : R, Rabs (f b - f a) = Rabs (f' c) * Rabs (b - a) /\
             Rmin a b <= c <= Rmax a b.
forall (f f' : R -> R) (a b : R),
(forall c : R,
 Rmin a b <= c <= Rmax a b ->
 derivable_pt_lim f c (f' c)) ->
exists c : R,
  Rabs (f b - f a) = Rabs (f' c) * Rabs (b - a) /\
  Rmin a b <= c <= Rmax a b
Proof.
forall (f f' : R -> R) (a b : R),
(forall c : R,
 Rmin a b <= c <= Rmax a b ->
 derivable_pt_lim f c (f' c)) ->
exists c : R,
  Rabs (f b - f a) = Rabs (f' c) * Rabs (b - a) /\
  Rmin a b <= c <= Rmax a b
intros f f' a b.
f, f':R -> R
a, b:R
(forall c : R,
 Rmin a b <= c <= Rmax a b ->
 derivable_pt_lim f c (f' c)) ->
exists c : R,
  Rabs (f b - f a) = Rabs (f' c) * Rabs (b - a) /\
  Rmin a b <= c <= Rmax a b
destruct (Rle_dec a b) as [aleb | blta].
f, f':R -> R
a, b:R
aleb:a <= b
(forall c : R,
 Rmin a b <= c <= Rmax a b ->
 derivable_pt_lim f c (f' c)) ->
exists c : R,
  Rabs (f b - f a) = Rabs (f' c) * Rabs (b - a) /\
  Rmin a b <= c <= Rmax a b
f, f':R -> R
a, b:R
blta:~ a <= b
(forall c : R,
 Rmin a b <= c <= Rmax a b ->
 derivable_pt_lim f c (f' c)) ->
exists c : R,
  Rabs (f b - f a) = Rabs (f' c) * Rabs (b - a) /\
  Rmin a b <= c <= Rmax a b
 destruct (Req_dec a b) as [ab | anb].
f, f':R -> R
a, b:R
aleb:a <= b
ab:a = b
(forall c : R,
 Rmin a b <= c <= Rmax a b ->
 derivable_pt_lim f c (f' c)) ->
exists c : R,
  Rabs (f b - f a) = Rabs (f' c) * Rabs (b - a) /\
  Rmin a b <= c <= Rmax a b
f, f':R -> R
a, b:R
aleb:a <= b
anb:a <> b
(forall c : R,
 Rmin a b <= c <= Rmax a b ->
 derivable_pt_lim f c (f' c)) ->
exists c : R,
  Rabs (f b - f a) = Rabs (f' c) * Rabs (b - a) /\
  Rmin a b <= c <= Rmax a b
f, f':R -> R
a, b:R
blta:~ a <= b
(forall c : R,
 Rmin a b <= c <= Rmax a b ->
 derivable_pt_lim f c (f' c)) ->
exists c : R,
  Rabs (f b - f a) = Rabs (f' c) * Rabs (b - a) /\
  Rmin a b <= c <= Rmax a b
  unfold Rminus; intros _; exists a; split.
f, f':R -> R
a, b:R
aleb:a <= b
ab:a = b
Rabs (f b + - f a) = Rabs (f' a) * Rabs (b + - a)
f, f':R -> R
a, b:R
aleb:a <= b
ab:a = b
Rmin a b <= a <= Rmax a b
f, f':R -> R
a, b:R
aleb:a <= b
anb:a <> b
(forall c : R,
 Rmin a b <= c <= Rmax a b ->
 derivable_pt_lim f c (f' c)) ->
exists c : R,
  Rabs (f b - f a) = Rabs (f' c) * Rabs (b - a) /\
  Rmin a b <= c <= Rmax a b
f, f':R -> R
a, b:R
blta:~ a <= b
(forall c : R,
 Rmin a b <= c <= Rmax a b ->
 derivable_pt_lim f c (f' c)) ->
exists c : R,
  Rabs (f b - f a) = Rabs (f' c) * Rabs (b - a) /\
  Rmin a b <= c <= Rmax a b
   now rewrite <- ab, !Rplus_opp_r, Rabs_R0, Rmult_0_r.
f, f':R -> R
a, b:R
aleb:a <= b
ab:a = b
Rmin a b <= a <= Rmax a b
f, f':R -> R
a, b:R
aleb:a <= b
anb:a <> b
(forall c : R,
 Rmin a b <= c <= Rmax a b ->
 derivable_pt_lim f c (f' c)) ->
exists c : R,
  Rabs (f b - f a) = Rabs (f' c) * Rabs (b - a) /\
  Rmin a b <= c <= Rmax a b
f, f':R -> R
a, b:R
blta:~ a <= b
(forall c : R,
 Rmin a b <= c <= Rmax a b ->
 derivable_pt_lim f c (f' c)) ->
exists c : R,
  Rabs (f b - f a) = Rabs (f' c) * Rabs (b - a) /\
  Rmin a b <= c <= Rmax a b
  split;[apply Rmin_l | apply Rmax_l].
f, f':R -> R
a, b:R
aleb:a <= b
anb:a <> b
(forall c : R,
 Rmin a b <= c <= Rmax a b ->
 derivable_pt_lim f c (f' c)) ->
exists c : R,
  Rabs (f b - f a) = Rabs (f' c) * Rabs (b - a) /\
  Rmin a b <= c <= Rmax a b
f, f':R -> R
a, b:R
blta:~ a <= b
(forall c : R,
 Rmin a b <= c <= Rmax a b ->
 derivable_pt_lim f c (f' c)) ->
exists c : R,
  Rabs (f b - f a) = Rabs (f' c) * Rabs (b - a) /\
  Rmin a b <= c <= Rmax a b
 rewrite Rmax_right, Rmin_left; auto; intros derv.
f, f':R -> R
a, b:R
aleb:a <= b
anb:a <> b
derv:forall c : R,
a <= c <= b -> derivable_pt_lim f c (f' c)
exists c : R,
  Rabs (f b - f a) = Rabs (f' c) * Rabs (b - a) /\
  a <= c <= b
f, f':R -> R
a, b:R
blta:~ a <= b
(forall c : R,
 Rmin a b <= c <= Rmax a b ->
 derivable_pt_lim f c (f' c)) ->
exists c : R,
  Rabs (f b - f a) = Rabs (f' c) * Rabs (b - a) /\
  Rmin a b <= c <= Rmax a b
 destruct (MVT_cor2 f f' a b) as [c [hc intc]];
  [destruct aleb;[assumption | contradiction] | apply derv | ].
f, f':R -> R
a, b:R
aleb:a <= b
anb:a <> b
derv:forall c0 : R,
a <= c0 <= b -> derivable_pt_lim f c0 (f' c0)
c:R
hc:f b - f a = f' c * (b - a)
intc:a < c < b
exists c0 : R,
  Rabs (f b - f a) = Rabs (f' c0) * Rabs (b - a) /\
  a <= c0 <= b
f, f':R -> R
a, b:R
blta:~ a <= b
(forall c : R,
 Rmin a b <= c <= Rmax a b ->
 derivable_pt_lim f c (f' c)) ->
exists c : R,
  Rabs (f b - f a) = Rabs (f' c) * Rabs (b - a) /\
  Rmin a b <= c <= Rmax a b
 exists c; rewrite hc, Rabs_mult;split;
  [reflexivity | unfold Rle; tauto].
f, f':R -> R
a, b:R
blta:~ a <= b
(forall c : R,
 Rmin a b <= c <= Rmax a b ->
 derivable_pt_lim f c (f' c)) ->
exists c : R,
  Rabs (f b - f a) = Rabs (f' c) * Rabs (b - a) /\
  Rmin a b <= c <= Rmax a b
assert (b < a) by (apply Rnot_le_gt; assumption).
f, f':R -> R
a, b:R
blta:~ a <= b
H:b < a
(forall c : R,
 Rmin a b <= c <= Rmax a b ->
 derivable_pt_lim f c (f' c)) ->
exists c : R,
  Rabs (f b - f a) = Rabs (f' c) * Rabs (b - a) /\
  Rmin a b <= c <= Rmax a b
assert (b <= a) by (apply Rlt_le; assumption).
f, f':R -> R
a, b:R
blta:~ a <= b
H:b < a
H0:b <= a
(forall c : R,
 Rmin a b <= c <= Rmax a b ->
 derivable_pt_lim f c (f' c)) ->
exists c : R,
  Rabs (f b - f a) = Rabs (f' c) * Rabs (b - a) /\
  Rmin a b <= c <= Rmax a b
rewrite Rmax_left, Rmin_right; try assumption; intros derv.
f, f':R -> R
a, b:R
blta:~ a <= b
H:b < a
H0:b <= a
derv:forall c : R,
b <= c <= a -> derivable_pt_lim f c (f' c)
exists c : R,
  Rabs (f b - f a) = Rabs (f' c) * Rabs (b - a) /\
  b <= c <= a
destruct (MVT_cor2 f f' b a) as [c [hc intc]];
 [assumption | apply derv | ].
f, f':R -> R
a, b:R
blta:~ a <= b
H:b < a
H0:b <= a
derv:forall c0 : R,
b <= c0 <= a -> derivable_pt_lim f c0 (f' c0)
c:R
hc:f a - f b = f' c * (a - b)
intc:b < c < a
exists c0 : R,
  Rabs (f b - f a) = Rabs (f' c0) * Rabs (b - a) /\
  b <= c0 <= a
exists c; rewrite <- Rabs_Ropp, Ropp_minus_distr, hc, Rabs_mult.
f, f':R -> R
a, b:R
blta:~ a <= b
H:b < a
H0:b <= a
derv:forall c0 : R,
b <= c0 <= a -> derivable_pt_lim f c0 (f' c0)
c:R
hc:f a - f b = f' c * (a - b)
intc:b < c < a
Rabs (f' c) * Rabs (a - b) =
Rabs (f' c) * Rabs (b - a) /\ 
b <= c <= a
split;[now rewrite <- (Rabs_Ropp (b - a)), Ropp_minus_distr| unfold Rle; tauto].
Qed.